Removes and replaces the built-in taxonomy metabox with our radio-select metabox.

taxonomy-radio-metabox.php

<?php

if ( !class_exists( 'WDS_Taxonomy_Radio' ) ) {
	/**
	 * Removes and replaces the built-in taxonomy metabox with our radio-select metabox.
	 * @link  http://codex.wordpress.org/Function_Reference/add_meta_box#Parameters
	 */
	class WDS_Taxonomy_Radio {

		// Post types where metabox should be replaced (defaults to all post_types associated with taxonomy)
		public $post_types = array();
		// Taxonomy slug
		public $slug = '';
		// Taxonomy object
		public $taxonomy = false;
		// New metabox title. Defaults to Taxonomy name
		public $metabox_title = '';
		// Metabox priority. (vertical placement)
		// 'high', 'core', 'default' or 'low'
		public $priority = 'high';
		// Metabox position. (column placement)
		// 'normal', 'advanced', or 'side'
		public $context = 'side';
		// Set to true to hide "None" option & force a term selection
		public $force_selection = false;


		/**
		 * Initiates our metabox action
		 * @param string $tax_slug      Taxonomy slug
		 * @param array  $post_types    post-types to display custom metabox
		 */
		public function __construct( $tax_slug, $post_types = array() ) {

			$this->slug = $tax_slug;
			$this->post_types = is_array( $post_types ) ? $post_types : array( $post_types );

			add_action( 'add_meta_boxes', array( $this, 'add_radio_box' ) );
		}

		/**
		 * Removes and replaces the built-in taxonomy metabox with our own.
		 */
		public function add_radio_box() {
			foreach ( $this->post_types() as $key => $cpt ) {
				// remove default category type metabox
				remove_meta_box( $this->slug .'div', $cpt, 'core' );
				// remove default tag type metabox
				remove_meta_box( 'tagsdiv-'.$this->slug, $cpt, 'core' );
				// add our custom radio box
				add_meta_box( $this->slug .'_radio', $this->metabox_title(), array( $this, 'radio_box' ), $cpt, $this->context, $this->priority );
			}
		}

		/**
		 * Displays our taxonomy radio box metabox
		 */
		public function radio_box() {

			// uses same noncename as default box so no save_post hook needed
			wp_nonce_field( 'taxonomy_'. $this->slug, 'taxonomy_noncename' );

			// get terms associated with this post
			$names = wp_get_object_terms( get_the_ID(), $this->slug );
			// get all terms in this taxonomy
			$terms = (array) get_terms( $this->slug, 'hide_empty=0' );
			// filter the ids out of the terms
			$existing = ( !is_wp_error( $names ) && !empty( $names ) )
				? (array) wp_list_pluck( $names, 'term_id' )
				: array();
			// Check if taxonomy is hierarchical
			// Terms are saved differently between types
			$h = $this->taxonomy()->hierarchical;

			// default value
			$default_val = $h ? 0 : '';
			// input name
			$name = $h ? 'tax_input['. $this->slug .'][]' : 'tax_input['. $this->slug .']';

			echo '<div style="margin-bottom: 5px;">
			<ul id="'. $this->slug .'_taxradiolist" data-wp-lists="list:'. $this->slug .'_tax" class="categorychecklist form-no-clear">';

				// If 'category,' force a selection, or force_selection is true
				if ( $this->slug != 'category' && !$this->force_selection ) {
					// our radio for selecting none
					echo '<li id="'. $this->slug .'_tax-0"><label><input value="'. $default_val .'" type="radio" name="'. $name .'" id="in-'. $this->slug .'_tax-0" ';
					checked( empty( $existing ) );
					echo '> '. sprintf( __( 'No %s', 'wds' ), $this->taxonomy()->labels->singular_name ) .'</label></li>';
				}

			// loop our terms and check if they're associated with this post
			foreach ( $terms as $term ) {

				$val = $h ? $term->term_id : $term->slug;

				echo '<li id="'. $this->slug .'_tax-'. $term->term_id .'"><label><input value="'. $val .'" type="radio" name="'. $name .'" id="in-'. $this->slug .'_tax-'. $term->term_id .'" ';
				// if so, they get "checked"
				checked( !empty( $existing ) && in_array( $term->term_id, $existing ) );
				echo '> '. $term->name .'</label></li>';
			}
			echo '</ul></div>';

		}

		/**
		 * Gets the taxonomy object from the slug
		 * @return object Taxonomy object
		 */
		public function taxonomy() {
			$this->taxonomy = $this->taxonomy ? $this->taxonomy : get_taxonomy( $this->slug );
			return $this->taxonomy;
		}

		/**
		 * Gets the taxonomy's associated post_types
		 * @return array Taxonomy's associated post_types
		 */
		public function post_types() {
			$this->post_types = !empty( $this->post_types ) ? $this->post_types : $this->taxonomy()->object_type;
			return $this->post_types;
		}

		/**
		 * Gets the metabox title from the taxonomy object's labels (or uses the passed in title)
		 * @return string Metabox title
		 */
		public function metabox_title() {
			$this->metabox_title = !empty( $this->metabox_title ) ? $this->metabox_title : $this->taxonomy()->labels->name;
			return $this->metabox_title;
		}


	}

	$custom_tax_mb = new WDS_Taxonomy_Radio( 'custom-tax-slug' );

	// Update optional properties

	// $custom_tax_mb->priority = 'low';
	// $custom_tax_mb->context = 'normal';
	// $custom_tax_mb->metabox_title = __( 'Custom Metabox Title', 'yourtheme' );
	// $custom_tax_mb->force_selection = true;
}

Some thoughts:

• Use the add_meta_boxes hook rather than admin_menu. The latter works, obviously, but it's always better to be semantic.
• Core saving of non-hierachical taxonomy terms expects a slug, not the ID. You'll end up creating new terms that are named things like "55" (or assigning to that term, if it exists already).
• Instead of blindly removing the metaboxes, use is_taxonomy_hierarchical() to check instead (also good for the above point).
• Radio for selecting none is not translatable. There's no good existing label to use, though. Personally, I wouldn't bother having this option, as I wouldn't use a radio select in a situation where you'd need to clear your selection.
• If you replace the category metabox, you can actually set no category. I'm not 100% sure on this, but I have a feeling there are places in core that assume that a category is always set for posts.
• I wouldn't place the metabox high on the side - by default for me it appeared above the publish metabox, which is probably not what you want.
• To help avoid conflicts, wrap this up in class_exists().
• Nitpicky semantics, but in the constructor, I'd put the $tax_slug arg first. You could even make the $post_types arg optional, then, and just have it do it for all post types with that taxonomy associated by default.

@helenhousandi Awesome pointers. Thanks for taking the time to look over the snippet & write your thoughts.

• Changed!
• I've updated the class to save the terms properly for both types (hierarchical and non)
• Similar solution used
• I made it a translatable string & also made that an optional class parameter
• Conditionally removed the "none" option for 'category'
• Yep, this is definitely a preference/need-based issue, so I've set the priority and the context as editable parameters
• Updated!
• I like it! Done!