This is the walker class that builds the proper checkbox list. It needs to be an include.

gistfile1.php

<?php
//
// Category Checklists
//

/**
* {@internal Missing Short Description}}
*
* @since 2.5.1
*/
if(!class_exists('MAE_Walker_Category_Checklist')){
class autoinc {
static $cnt=0;
static function increment() {
return self::$cnt++;
}
static function reset() {
self::$cnt=0;
}
}
class MAE_Walker_Category_Checklist extends Walker {
var $tree_type = 'category';
var $db_fields = array ('parent' => 'parent', 'id' => 'term_id'); //TODO: decouple this

function start_lvl(&$output, $depth, $args) {
$indent = str_repeat("\t", $depth);
$output .= "$indent<ul class='children'>\n";
//$output .= "$indent\n";
}

function end_lvl(&$output, $depth, $args) {
$indent = str_repeat("\t", $depth);
$output .= "$indent</ul>\n";
//$output .= "$indent\n";
}

function start_el(&$output, $category, $depth, $args) {
extract($args);
if ( empty($taxonomy) )
$taxonomy = 'category';

if ( $taxonomy == 'category' )
$name = 'post_category';
else
$name = $taxonomy;
//$name = 'tax_input['.$taxonomy.']';
$index = autoinc::increment();
// d($args);
$class = in_array( $category->term_id, $popular_cats ) ? ' class="popular-category"' : '';
//$output .= "\n<li id='{$taxonomy}-{$category->term_id}'$class>" . '<label class="selectit"><input value="' . $category->term_id . '" type="checkbox" name="'.$name.'[]" id="in-'.$taxonomy.'-' . $category->term_id . '"' . checked( in_array( $category->term_id, $selected_cats ), true, false ) . disabled( empty( $args['disabled'] ), false, false ) . ' /> ' . esc_html( apply_filters('the_category', $category->name )) . '</label>';
$output .= "\n<li id='{$taxonomy}-{$category->term_id}'$class>" . '<label class="piklist-field-list-item '.(isset($args['attclass']) ? implode(' ', $args['attclass']) : '').'"><input value="' . $category->term_id . '" type="checkbox" name="'.piklist_form::get_field_name($args['pl_field'], $args['pl_scope'], $index, $args['pl_prefix']).'" id="'.piklist_form::get_field_id($args['pl_field'], $args['pl_scope'], $index, $args['pl_prefix']).'"' . checked( in_array( $category->term_id, $selected_cats ), true, false ) . disabled( empty( $args['disabled'] ), false, false ) . ' /> ' . esc_html( apply_filters('the_category', $category->name )) . '</label>';
//$output .= '<label class="fullsize customcheckboxl" for="'.esc_html( apply_filters('the_category', $category->name )).$category->term_id.'"><input type="checkbox" id="'.esc_html( apply_filters('the_category', $category->name )).$category->term_id.'" class="customcheckbox" name="'.$name.'[]" '.checked( in_array( $category->term_id, $selected_cats ), true, false ).' value="'.$category->term_id.'" /> '.esc_html( apply_filters('the_category', $category->name )).'</label>';
//'<label class="fullsize customcheckboxl" for="'.$language_list->cat_name.$language_list->cat_ID.'"><input type="checkbox" id="'.$language_list->cat_name.$language_list->cat_ID.'" class="customcheckbox" name="language[]" '.$check.' value="'.$language_list->cat_ID.'" /> '.$language_list->cat_name.'</label>';
}

function end_el(&$output, $category, $depth, $args) {
$output .= "</li>\n";
//$output .= "\n";
}
}
}
/**
* {@internal Missing Short Description}}
*
* @since 2.5.1
*
* @param unknown_type $post_id
* @param unknown_type $descendants_and_self
* @param unknown_type $selected_cats
* @param unknown_type $popular_cats
*/
if(!function_exists('mae_category_checklist')){
function mae_category_checklist( $post_id = 0, $descendants_and_self = 0, $selected_cats = false, $popular_cats = false, $walker = null, $checked_ontop = true ) {
mae_terms_checklist($post_id,
array(
'taxonomy' => 'category',
'descendants_and_self' => $descendants_and_self,
'selected_cats' => $selected_cats,
'popular_cats' => $popular_cats,
'walker' => $walker,
'checked_ontop' => $checked_ontop
 ));
}
}
/**
* Taxonomy independent version of mae_category_checklist
*
* @since 3.0.0
*
* @param int $post_id
* @param array $args
*/
if(!function_exists('mae_terms_checklist')){
function mae_terms_checklist($post_id = 0, $args = array()) {
$defaults = array(
'descendants_and_self' => 0,
'selected_cats' => false,
'popular_cats' => false,
'walker' => null,
'taxonomy' => 'category',
'checked_ontop' => false
);
extract( wp_parse_args($args, $defaults), EXTR_SKIP );

if ( empty($walker) || !is_a($walker, 'Walker') )
$walker = new MAE_Walker_Category_Checklist;

$descendants_and_self = (int) $descendants_and_self;

$args = array('taxonomy' => $taxonomy);

$tax = get_taxonomy($taxonomy);
$args['disabled'] = !current_user_can($tax->cap->assign_terms);

if ( is_array( $selected_cats ) )
$args['selected_cats'] = $selected_cats;
elseif ( $post_id )
$args['selected_cats'] = wp_get_object_terms($post_id, $taxonomy, array_merge($args, array('fields' => 'ids')));
else
$args['selected_cats'] = array();

if ( is_array( $popular_cats ) )
$args['popular_cats'] = $popular_cats;
else
$args['popular_cats'] = get_terms( $taxonomy, array( 'fields' => 'ids', 'orderby' => 'count', 'order' => 'DESC', 'number' => 10, 'hierarchical' => false ) );

if ( $descendants_and_self ) {
$categories = (array) get_terms($taxonomy, array( 'child_of' => $descendants_and_self, 'hierarchical' => 0, 'hide_empty' => 0 ) );
$self = get_term( $descendants_and_self, $taxonomy );
array_unshift( $categories, $self );
} else {
$categories = (array) get_terms($taxonomy, array('get' => 'all'));
}
$args['pl_field']=$pl_field;
$args['pl_scope']=$pl_scope;
$args['pl_prefix']=$pl_prefix;
$args['attclass']=$attclass;

if ( $checked_ontop ) {
// Post process $categories rather than adding an exclude to the get_terms() query to keep the query the same across all posts (for any query cache)
$checked_categories = array();
$keys = array_keys( $categories );

foreach( $keys as $k ) {
if ( in_array( $categories[$k]->term_id, $args['selected_cats'] ) ) {
$checked_categories[] = $categories[$k];
unset( $categories[$k] );
}
}

// Put checked cats on top
echo call_user_func_array(array(&$walker, 'walk'), array($checked_categories, 0, $args));
}
// Then the rest of them
echo call_user_func_array(array(&$walker, 'walk'), array($categories, 0, $args));
autoinc::reset();
}
}

//
// Category RadioLists
//

/* USAGE:
<ul class="nonbulletlist">
    <?php
       wp_terms_radiolist($postid, array( 'taxonomy' => 'category') );
     ?>
</ul>

*/

/**
* {@internal Missing Short Description}}
*
* @since 2.5.1
*/
if(!class_exists('Walker_Category_Radiolist')){
class Walker_Category_Radiolist extends Walker {
   var $tree_type = 'category';
   var $db_fields = array ('parent' => 'parent', 'id' => 'term_id'); //TODO: decouple this

   function start_lvl(&$output, $depth, $args) {
       $indent = str_repeat("\t", $depth);
       $output .= "$indent<ul class='children'>\n";
               //$output .= "$indent\n";
   }

   function end_lvl(&$output, $depth, $args) {
       $indent = str_repeat("\t", $depth);
       $output .= "$indent</ul>\n";
               //$output .= "$indent\n";
   }

   function start_el(&$output, $category, $depth, $args) {
       extract($args);
       if ( empty($taxonomy) )
           $taxonomy = 'category';

       if ( $taxonomy == 'category' )
           $name = 'post_category';
       else
                       $name = $taxonomy;
           //$name = 'tax_input['.$taxonomy.']';

       $class = in_array( $category->term_id, $popular_cats ) ? ' class="popular-category"' : '';
       //$output .= "\n<li id='{$taxonomy}-{$category->term_id}'$class>" . '<label class="selectit"><input value="' . $category->term_id . '" type="checkbox" name="'.$name.'[]" id="in-'.$taxonomy.'-' . $category->term_id . '"' . checked( in_array( $category->term_id, $selected_cats ), true, false ) . disabled( empty( $args['disabled'] ), false, false ) . ' /> ' . esc_html( apply_filters('the_category', $category->name )) . '</label>';
               $output .= "\n<li id='{$taxonomy}-{$category->term_id}'$class>" . '<label class="selectit"><input value="' . $category->term_id . '" type="radio" name="'.$name.'[]" id="in-'.$taxonomy.'-' . $category->term_id . '"' . checked( in_array( $category->term_id, $selected_cats ), true, false ) . disabled( empty( $args['disabled'] ), false, false ) . ' /> ' . esc_html( apply_filters('the_category', $category->name )) . '</label>';
               //$output .= '<label class="fullsize customcheckboxl" for="'.esc_html( apply_filters('the_category', $category->name )).$category->term_id.'"><input type="checkbox" id="'.esc_html( apply_filters('the_category', $category->name )).$category->term_id.'" class="customcheckbox" name="'.$name.'[]" '.checked( in_array( $category->term_id, $selected_cats ), true, false ).' value="'.$category->term_id.'" /> '.esc_html( apply_filters('the_category', $category->name )).'</label>';
               //'<label class="fullsize customcheckboxl" for="'.$language_list->cat_name.$language_list->cat_ID.'"><input type="checkbox" id="'.$language_list->cat_name.$language_list->cat_ID.'" class="customcheckbox" name="language[]" '.$check.' value="'.$language_list->cat_ID.'" /> '.$language_list->cat_name.'</label>';
   }

   function end_el(&$output, $category, $depth, $args) {
       $output .= "</li>\n";
               //$output .= "\n";
   }
}
}
/**
* {@internal Missing Short Description}}
*
* @since 2.5.1
*
* @param unknown_type $post_id
* @param unknown_type $descendants_and_self
* @param unknown_type $selected_cats
* @param unknown_type $popular_cats
*/
if(!function_exists('wp_category_radiolist')){
function wp_category_radiolist( $post_id = 0, $descendants_and_self = 0, $selected_cats = false, $popular_cats = false, $walker = null, $checked_ontop = true ) {
       wp_terms_radiolist($post_id,
           array(
                   'taxonomy' => 'category',
                   'descendants_and_self' => $descendants_and_self,
                   'selected_cats' => $selected_cats,
                   'popular_cats' => $popular_cats,
                   'walker' => $walker,
                   'checked_ontop' => $checked_ontop
       ));
   }
}
/**
* Taxonomy independent version of mae_category_checklist
*
* @since 3.0.0
*
* @param int $post_id
* @param array $args
*/
if(!function_exists('wp_terms_radiolist')){
function wp_terms_radiolist($post_id = 0, $args = array()) {
   $defaults = array(
       'descendants_and_self' => 0,
       'selected_cats' => false,
       'popular_cats' => false,
       'walker' => null,
       'taxonomy' => 'category',
       'checked_ontop' => false
   );
   extract( wp_parse_args($args, $defaults), EXTR_SKIP );

   if ( empty($walker) || !is_a($walker, 'Walker') )
       $walker = new Walker_Category_Radiolist;

   $descendants_and_self = (int) $descendants_and_self;

   $args = array('taxonomy' => $taxonomy);

   $tax = get_taxonomy($taxonomy);
   $args['disabled'] = !current_user_can($tax->cap->assign_terms);

   if ( is_array( $selected_cats ) )
       $args['selected_cats'] = $selected_cats;
   elseif ( $post_id )
       $args['selected_cats'] = wp_get_object_terms($post_id, $taxonomy, array_merge($args, array('fields' => 'ids')));
   else
       $args['selected_cats'] = array();

   if ( is_array( $popular_cats ) )
       $args['popular_cats'] = $popular_cats;
   else
       $args['popular_cats'] = get_terms( $taxonomy, array( 'fields' => 'ids', 'orderby' => 'count', 'order' => 'DESC', 'number' => 10, 'hierarchical' => false ) );

   if ( $descendants_and_self ) {
       $categories = (array) get_terms($taxonomy, array( 'child_of' => $descendants_and_self, 'hierarchical' => 0, 'hide_empty' => 0 ) );
       $self = get_term( $descendants_and_self, $taxonomy );
       array_unshift( $categories, $self );
   } else {
       $categories = (array) get_terms($taxonomy, array('get' => 'all'));
   }

   if ( $checked_ontop ) {
       // Post process $categories rather than adding an exclude to the get_terms() query to keep the query the same across all posts (for any query cache)
       $checked_categories = array();
       $keys = array_keys( $categories );

       foreach( $keys as $k ) {
           if ( in_array( $categories[$k]->term_id, $args['selected_cats'] ) ) {
               $checked_categories[] = $categories[$k];
               unset( $categories[$k] );
           }
       }

       // Put checked cats on top
       echo call_user_func_array(array(&$walker, 'walk'), array($checked_categories, 0, $args));
   }
   // Then the rest of them
   echo call_user_func_array(array(&$walker, 'walk'), array($categories, 0, $args));
}
}

?>