Application stopped working -- claims it requires new OS X version, but it worked before
According to Updating GitHub Desktop the client updates itself automatically:


  GitHub Desktop automatically downloads updates and installs them when you restart. You can also manually check for updates.


It appears that Github has updated their build toolchain server to a newer version of Xcode not supporting Lion.

You need to stay on the old version.  

You may want to contact Github to get the newest build you can still use.