How to handle a post request from an external application in Drupal?
Your path is wrong I think. You should be able to access that callback here.

https://mydrupalsite.com/bh_commerce_stripe_oauth/callback/randomstring

So actually your hook_menu should look like this.

/* Implements hook_menu */
function bh_commerce_stripe_oauth_menu() {
  $items = array();
  $items['callback'] = array(
    'title' =&gt; 'bh_commerce_stripe_oauth_callback',
    'page callback' =&gt; '_bh_commerce_stripe_oauth_handle_callback',
    'type' =&gt; MENU_CALLBACK     
  );
  return $items;    
}


Note: You don't have any path wildcards (represented by % in your array key) in the callback you are getting that are useful. Therefore you don't need any page arguments. The interesting things are in the $_GET (or $_REQUEST).

And then to access the results something like:

function _bh_commerce_stripe_oauth_handle_callback() {
  $scope = $_GET['scope'];
  $code = $_GET['code'];
  drupal_set_message(t('_commerce_stripe_oauth_handle_callback: called'));
  drupal_goto('&lt;front&gt;');
}