Well that's obviously not good. There's very little that actually changed in this release that should affect JomSocial's homepage (or really any extension). If you said v4.3.0 caused issues, I'd be less surprised since that had some sweeping changes.
So.. if it v4.3.0 works, that does help narrow some ideas down. However, can you let me know a little more about what "stopped working"? Did you get a blank page, an error, or the Login Button simply didn't work (no popup, error, etc)? If it was a blank page, can you set Error Reporting in Joomla's Global Config to Maximum and see if there's a pretty PHP error message? If so, post that, as it will help us narrow down the problem.
Beyond that, v4.3.0 has no major bugs, v4.3.1 just has some minor improvements. Sticking with v4.3.0 until we figure out what's happening should be fine. We haven't made it permanent, but I just updated our demo site to v4.3.1 and the JomSocial home page works fine there. If you check before it resets itself in a few hours, you can see it working there (not that that's very exciting..).
Thanks,
Alex