Updated Instagram Integration with Appcelerator, all the files you need are here, be sure to include the promises library in your project

credentials.js

exports.C = {
    INSTAGRAM_CLIENT_ID : 'replace this with your public', //replace this with your public
    INSTAGRAM_CLIENT_SECRET : 'replace this with your private key', //replace this with your private key
    INSTAGRAM_CALLBACK_URL : 'replace this with your INSTAGRAM_CALLBACK_URL'
};

index.js

    var creds = require('credentials').C;
    var instagramMgr = require('instagram-module');
    var client = new instagramMgr.Client(creds.INSTAGRAM_CLIENT_ID, 
                                         creds.INSTAGRAM_CLIENT_SECRET, 
                                         creds.INSTAGRAM_CALLBACK_URL);
    client.login().then(function(_result) {
        Ti.API.info('_result ' + JSON.stringify(_result));
    }, function error(_error) {
        Ti.API.info('_error ' + JSON.stringify(_error));
    });

instagram-module.js

//Public client interface
function Client(clientId, clientSecret, callbackUrl) {
    this.INSTAGRAM_CLIENT_ID = clientId;
    this.INSTAGRAM_CLIENT_SECRET = clientSecret;
    this.INSTAGRAM_CALLBACK_URL = callbackUrl;
    this.ENDPOINT = 'https://api.instagram.com/v1/';
    this.ACCESS_TOKEN = null;
    this.LOGIN_PROMISE = null;

    Ti.API.debug(JSON.stringify(this));

};

exports.Client = Client;

var window, webView, success_callback;

// REMEMBER TO INCLUDE THIS LIBRARY IN PROJECT https://github.com/kriskowal/q
var Q = require('q'); 

/**
 * @param authSuccess_callback
 * @param _scope
 */
Client.prototype.login = function(authSuccess_callback, _scope) {
    var deferred = Q.defer();

    var scope, that = this;
    scope = _scope || "likes+comments";
    that.LOGIN_PROMISE = deferred;

    if (Ti.App.Properties.getString("INSTAGRAM.ACCESS_TOKEN") === null) {
        if (authSuccess_callback !== undefined) {
            success_callback = authSuccess_callback;
        }
        showAuthorizeUI(String.format('https://instagram.com/oauth/authorize/?response_type=token&client_id=%s&redirect_uri=%s&scope=%s&display=touch', that.INSTAGRAM_CLIENT_ID, that.INSTAGRAM_CALLBACK_URL, scope), that);
    } else {
        Ti.API.info('login: using saved token');
        that.ACCESS_TOKEN = Ti.App.Properties.getString("INSTAGRAM.ACCESS_TOKEN");

        if (that.ACCESS_TOKEN) {
            // call the callback since we were successful
            var results = {
                access_token : that.ACCESS_TOKEN
            };

            authSuccess_callback && success_callback(result);
            deferred.resolve(result);

            Titanium.App.Properties.removeProperty("INSTAGRAM.ACCESS_TOKEN");
        }
    }
    return deferred.promise;
};
/**
 * code to display the familiar web login dialog we all know and love
 *
 * @param pUrl
 * @param _that -  the context of this instance of this module
 */
function showAuthorizeUI(pUrl, _that) {
    var that = _that;

    window = Ti.UI.createWindow({
        top : 0,
        //modal : true,
        fullscreen : true
    });
    webView = Ti.UI.createWebView({
        url : pUrl,
        scalesPageToFit : true,
        touchEnabled : true,
        top : 0,
        backgroundColor : '#FFF'
    });
    Ti.API.debug('Setting:[' + Ti.UI.AUTODETECT_NONE + ']');
    webView.addEventListener('beforeload', function(e) {
        if (e.url.indexOf(that.INSTAGRAM_CALLBACK_URL) != -1 || e.url.indexOf('instagram') != -1) {
            Titanium.API.debug('in before load ' + JSON.stringify(e));
            authorizeUICallback(e, that.LOGIN_PROMISE);
            webView.stopLoading = true;
        }
    });
    webView.addEventListener('load', function(e) {
        authorizeUICallback(e, that.LOGIN_PROMISE);
    });
    window.add(webView);
    window.open();

};

/**
 * unloads the UI used to have the user authorize the application
 */
function destroyAuthorizeUI() {
    Ti.API.debug('destroyAuthorizeUI');
    // if the window doesn't exist, exit
    if (window == null) {
        return;
    }

    // remove the UI
    try {
        Ti.API.debug('destroyAuthorizeUI:webView.removeEventListener');
        webView.removeEventListener('load', authorizeUICallback);
        Ti.API.debug('destroyAuthorizeUI:window.close()');
        window.close();
    } catch(ex) {
        Ti.API.debug('Cannot destroy the authorize UI. Ignoring.');
    }
};

/**
 * fires event when login fails
 * <code>app:4square_access_denied</code>
 *
 * fires event when login successful
 * <code>app:4square_token</code>
 *
 * executes callback if specified when creating object
 */
function authorizeUICallback(e, _loginPromise) {
    Ti.API.debug('authorizeUILoaded ' + e.url);
    Titanium.API.debug("authorizeUICallback " + JSON.stringify(e));
    Titanium.API.debug("e.url " + e.url);

debugger;
    if (e.url && e.url.indexOf('#access_token') != -1) {
        var token = e.url.split("=")[1];
        this.ACCESS_TOKEN = token;
        Ti.App.fireEvent('app:instagram_token', {
            "data" : token
        });

        if (success_callback !== undefined) {
            success_callback({
                access_token : token
            });
        }

        _loginPromise.resolve({
            access_token : token
        });

        Titanium.API.debug("Saved Access Token " + token);

        destroyAuthorizeUI();

    } else if ('http://instagram.com/' == e.url) {
        Ti.App.fireEvent('app:instagram_logout', {});
        destroyAuthorizeUI();
        _loginPromise.resolve({
            access_token : null,
            logout : true
        });
    } else if (e.url.indexOf('#error=access_denied') != -1) {
        Ti.App.fireEvent('app:instagram_access_denied', {});
        destroyAuthorizeUI();
        _loginPromise.resolve({
            access_token : null,
            access_denied : true
        });
    }

};

/*
 * Get a Instagram object.  Argument hash:
 *
 * className: string name of the Instagram class
 * objectId: The object ID of the object you're getting, no id will return all
 * success: function to call on success
 * error: function to call on error
 *
 */
Client.prototype.get = function(args) {
    Ti.API.debug("Client.prototype.get: started");
    // get the login information and let it roll!!
    var url, that = this;

    try {

        if (that.xhr == null) {
            that.xhr = Titanium.Network.createHTTPClient();
            that.xhr.autoEncodeUrl = false;
        }

        if (that.ACCESS_TOKEN != null) {
            url = that.ENDPOINT + args.path + "?access_token=" + that.ACCESS_TOKEN;
        } else {
            url = that.ENDPOINT + args.path + "?client_id=" + that.CLIENT_ID;
        }
        for (var x = 0; x < args.params.length; x++) {
            url = url + "&" + args.params[x][0] + "=" + args.params[x][1];
        }
        Ti.API.debug("Client.prototype.get " + url);
        that.xhr.open("GET", url);

        that.xhr.onerror = function(e) {
            Ti.API.error("InstagramMgr ERROR " + e.error);
            Ti.API.error("InstagramMgr ERROR " + that.xhr.location);
            if (args.error) {
                args.error(e);
            }
        };

        that.xhr.onload = function(_xhr) {
            Ti.API.debug("InstagramMgr response: " + that.xhr.responseText);
            if (args.success) {
                args.success(that.xhr.responseText);
            }
        };

        that.xhr.send();
    } catch(err) {
        Titanium.UI.createAlertDialog({
            title : "Error",
            message : String(err),
            buttonNames : ['OK']
        }).show();
    }
};
/*
 * Get a Instagram object.  Argument hash:
 *
 * className: string name of the Instagram class
 * objectId: The object ID of the object you're getting, no id will return all
 * success: function to call on success
 * error: function to call on error
 *
 */
Client.prototype.post = function(args) {
    Ti.API.debug("Client.prototype.post: started");
    // get the login information and let it roll!!
    var url, that = this;

    try {

        if (that.xhr == null) {
            that.xhr = Titanium.Network.createHTTPClient();
            that.xhr.autoEncodeUrl = false;
        }

        if (that.ACCESS_TOKEN != null) {
            url = that.ENDPOINT + args.path + "?access_token=" + that.ACCESS_TOKEN;
        } else {
            url = that.ENDPOINT + args.path + "?client_id=" + that.CLIENT_ID;
        }

        Ti.API.debug("Client.prototype.post " + url);
        that.xhr.open("POST", url);

        that.xhr.onerror = function(e) {
            Ti.API.error("InstagramMgr ERROR " + e.error);
            Ti.API.error("InstagramMgr ERROR " + that.xhr.location);
            if (args.error) {
                args.error(e);
            }
        };

        that.xhr.onload = function(_xhr) {
            Ti.API.debug("InstagramMgr response: " + that.xhr.responseText);
            if (args.success) {
                args.success(that.xhr.responseText);
            }
        };

        that.xhr.send(args.params);
    } catch(err) {
        Titanium.UI.createAlertDialog({
            title : "Error",
            message : String(err),
            buttonNames : ['OK']
        }).show();
    }
};