How to create simple confirmation page?

I wrote a blog post that you may find helpful. Good luck!

function custom_menu() {
  $items = array();
  $items['admin/sendmail/%'] = array(
    'title' => t('Delete Entry'),
    'page callback' => 'drupal_get_form',
    'page arguments' => array('custom_confirm_form'),
    'access arguments' => array('send mail'),
    'type' => MENU_CALLBACK,
  );
  return $items;
}

  function custom_confirm_form($form_state){
    return confirm_form($form, t('Are you sure you want to Send Mail?'), //message title
    'admin/send-mail', //redirect if cancelled
    t('Only say yes if you are really sure. Otherwise please click on cancel button.'), //message description
    t('Yes'), //confirm button text
    t('Cancel') //cancel button text
  );
}

function custom_confirm_form_submit($form, &$form_state){
   //verify that the form was confirmed
  if ($form_state['values']['confirm']) {  
     custom_send_mail();
  }
}

I suggest you use hook_user_update. Then you compare $account->mail and $account->original->mail. If the two email addresses are different, the user's email address changed. Next you generate the token and send an email to the user with a link to your page.

For the page, create a new menu entry via hook_menu and do the logic for the REST call there. You can get URL query parameters (especially the token) by using drupal_get_query_parameters. Maybe you should ensure that the token is unique and store it with the email address and the creation time (to check if it has expired) in the database.

The documentation for hook_menu (or 99% of all other modules available) should give you an example on how to use it.