Same as shipping checkbox in admin is not showing
I'm guessing this checkbox is meant to copy the address from the billing address to the shipping address?

As Ryan pointed out on the frontend this will require ajax as the shipping and billing profiles could be on different steps, however on the backend they are both on the same page so all that is required is some javascript.

For example...

Add the checkbox and custom javascript to the page in a hook_form_alter:

/**
 * Implements hook_form_FORM_ID_alter() for commerce_order_ui_order_form.
 */
function MY_MODULE_form_commerce_order_ui_order_form_alter(&amp;$form, &amp;$form_state, $form_id) {

  // Add the checkbox
  $form['commerce_customer_billing'][LANGUAGE_NONE]['profiles'][0]['copy_from_shipping'] = [
    '#type' =&gt; 'checkbox',
    '#title' =&gt; t('Copy address from shipping'),
    '#attributes' =&gt; ['class' =&gt; ['copy-address-from-shipping']],
  ];

  // Attach our custom js
  $form['#attached']['js'][] = drupal_get_path('MY_MODULE', 'taronga') . '/js/copy-address-from-shipping.js';

}


And then for the javascript:

(function($) {

    /**
     * The Drupal behaviors to copy address from shipping to billing on admin order edit page.
     */
    Drupal.behaviors.copyShippingAddress = {
        attach: function(context) {

            // Only bind our click event handler once
            $('.copy-address-from-shipping').once(function(){

                // Bind the click event
                $('.copy-address-from-shipping').click(function () {

                    var $shipping = $('#edit-commerce-customer-shipping-und-profiles-0');
                    var $billing = $('#edit-commerce-customer-billing-und-profiles-0');

                    if($(this).is(':checked')) { // Copy values from shipping

                        // Loop over inputs &amp; select in shipping fieldset
                        $shipping.find('input, select').each(function () {
                            // Skip any inputs without a name attribute (e.g. chosen)
                            if($(this).is('[name]')) {
                                // Get the name of this field and convert it to the billing name
                                var billing_name = $(this).attr('name').replace('shipping', 'billing');
                                var $billing_field = $billing.find('[name="' + billing_name + '"]');
                                // Make sure the billing field exists
                                if($billing_field.length) {
                                    // Replace the value
                                    $billing_field.val($(this).val());
                                }
                            }
                        });

                    }
                    else { // Clear values

                        $billing.find('input, select').each(function(){
                            $(this).val('');
                        });

                    }
                });

            });

        }
    };

})(jQuery);