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This is the code you need. In the Drupal site create the following module:

Update:

function my_special_login_menu()
{
    $items['authinticate-by-customerid'] = array( // or any other custom path you choose
        'title' => 'Login by customerid',
        'access callback' => true, // no point in authenticating at this point in code execution
        'page callback' => 'my_special_login_function',
        'type' => MENU_CALLBACK,
        );
    return $items;
}

function my_special_login_function($customerid)
{
    // post the data to /authinticate-by-customerid as json with a parameter / property named customerid holding the customerid
    $postData = file_get_contents('php://input');

    $Data = json_decode($postData);

    $query = new EntityFieldQuery();
    $query->entityCondition('entity_type', 'user')
        ->propertyCondition('uid', 1, '<>')
        // User number 1 should not be able to log in using this method
        ->fieldCondition('field_customerid', 'value', $Data->customerid, '=')
        ->range(0,1);

    $result = $query->execute();

    if (isset($result['user'])) {
        $uid = array_keys($result['user'])[0];
        $account = user_load($uid);
        global $user;
        $user->uid = $uid;
        $user->name = $account->name;
        $user->timezone = $account->timezone;
        user_login_finalize(); // Log user in
        drupal_goto('/user'); // or any other path to redirect logged in users to
    } else {
        drupal_goto('/user/register');
    }

}

For this code to work you need Entity API module and you should serve the site over SSL in order to encrypt the data transferred between the two sites.