wp_ajax action responds with 0

Change your wp_localize_script-Call to the right Action:

wp_localize_script( 'addItemToNav', 'menuItems', array(
    // URL to wp-admin/admin-ajax.php to process the request
    'ajaxurl'          => admin_url( 'admin-ajax.php' ),
    // generate a nonce with a unique ID "myajax-post-comment-nonce"
    'postCommentNonce' => wp_create_nonce( 'update_nav_items-nonce' ),
    'action' => 'wp_ajax_update_nav_items'
    )
);

As ‘action’ you currently Input the function Name, but that is wrong. Your registered Action to call the function has an different Name.