Ahhhh! The issue described there seems to be very likely the cause.. or at least definitely a part of it. Both of your sites are using JomSocial as well.
We actually just tested on a JomSocial 3.1 installation and were noticing the logging out issues. It only happened when we navigated to a page that loaded JomSocial (either a part of it in a module or a main component page). Other pages kept us logged in just fine. We were just starting to debug to see if it was something in JFBConnect or related to your issues above. Guess I should have checked their blog first
Please keep me posted if that fixes the problem you're having as we're very curious as well. We'll be updating to JomSocial 3.2 shortly to test and see what happens as well, but would love to hear your feedback on developed sites.
Thanks!
Alex