It's not JFBConnect that would be causing that. Joomla stores a cookie in the user's browser that expires after whatever your Session Timeout setting is in the Global Configuration area of Joomla. If you also have your browser setup in a very secure manner, the cookies may be deleted every time you close. If that's the case, Joomla won't see that cookie and won't know who's logged in.
So, try with different browsers and configurations, but there's nothing in JFBConnect that should be doing what your saying.
Hope that helps explain,
Alex
Why does it log you out when closing browser
