Blogs MU custom login stopped working – get a "The connection was reset" error

The custom login on the top right of the header has stopped working. All users get a “The connection was reset. The connection to the server was reset while the page was loading.” error.

This is the login link that calls a page with a custom page template called “page-template-custom-login.php”. It had been working for us for many months, and now it has stopped working.

The login link in the Meta section, which calls wp-login directly, works perfectly.

We have cleared all cache both server and client side; disabled all plugins; enabled and disabled BuddyPress; reset Permalinks and nothing has resolved the issue.

We recently had .htaccess and permalinks issues which appear to have been resolved with the help of our host, WPEngine. During that process we reset our .htaccess file to be the standard recommended file provided with WordPress.

Any ideas about what could be causing this?