Manage Groups filters broken in 4.7.7?
Thanks for bringing this to a wider audience!  I just confirmed the issue on the demo site, which is the first step toward determining if this is a bug for everyone (see Reporting a Bug).

I've confirmed it's not an AJAX error or a problem with how the URL is constructed, which were my first guesses.  I'll take a few minutes to poke around in the code, and if the problem's not obvious I'll report it upstream at https://issues.civicrm.org.

UPDATE: The problem's not obvious to me, but I know there are some bugfixes in process for this set of new functionality (CRM-16353)