I don't have an idea what would have happened. JFBConnect doesn't just change on its own, so it's a little difficult to guess what did actually change. What I'd recommend, if the problem happens again, is going to the Joomla Global Configuration area. In there, set the Error Reporting option to 'Maximum'. Then, when you try to authenticate with Facebook, if you get a white page, there will hopefully be a full message detailing the problem.
With that, we can help investigate what could be the problem.
Re-installing the extension and it working means that some file may have been corrupt or was never installed properly in the first place. I'm glad to hear this worked for you, but not sure why it would have been necessary.
As for Autotune, that warning will come up whenever you haven't go through the settings in about 30-days. Please go into Autotune, verify your Subscriber ID (which you can get from any of the My Account menu links in the top right) and go through the FB Application and Error Check settings. Facebook changes frequently, and we update our checks and recommendations accordingly. If you don't run Autotune, JFBConnect will continue working just fine. We don't intentionally 'break' after a set time or anything like that.
I hope that helps get you going, but if you have any other questions, just let me know.
Thanks,
Alex