Same to me: after upgrading AirPort Express (AE) to 7.6.1 a few weeks ago, it regurlaly hangs, but works fine for hours or days in between. The AE extends a wireless network created by a Time Capsule (TC). TC was upgraded to 7.6.1 at the same time and works without problems.
When AE hangs, it's restarting itself all the time: the status light is solid amber for about a minute (start-up), then flashes two times amber (connecting), becomes green (ok) for half a second, and then solid amber again (restart). This cyclic restarting can last for hours. Very strange behavior... During such cyclic restarting, it is on so shortly that it never shows up in AirPort Utility and it is impossible to connect and read the log file.
Resetting to factory values and reconfiguring does not solve the problem. Configuring to join (instead of extend) the wireless network does not solve the problem. My next try will be to revert to firmware 7.6 (AirPort Utility 5.6 > Base Station > Upoad Firmware... > select previous version from the list).
Any other ideas around?