Something is altering our URLs and causing the issues you're seeing. When being redirected from Facebook, the user is being sent to:
/index.php?option=com_jfbconnect
That URL is missing a lot of required parameters. Can you let me know:
* Are you running any security extensions on your site? I
asked a similar question here
for an issue that may be related
* Can you disable any SEF extensions or settings you may have enabled, as those could be altering our URLs. If that works, we can help you re-enable them.
* Is there anything special on your site that may be causing alterations to URLs or redirections in general?
Hopefully the above gets you started. We'll need to narrow down the cause though to get things figured out.
Thanks,
Alex