There was a bug with Facebook's Javascript login button a while ago that had the same behavior as what you described. We haven't seen that issue in a few months, and unfortunately, there's not much we can do if it's back to fix it directly.
Can you let me know what type of logout you have in the main component? Specifically, when logging out, is the user logging out of your Joomla site only, or both Facebook and your Joomla site? I'd try switching it to logout of Facebook which may fix the issue (though may not be what you're looking for).
The other best recommendation I have is to use the image-type button available in the SCLogin module or that can be implemented with some HTML. Unfortunately though, the image button doesn't have the ability to show faces of friends that use the app.
Beyond that, if it's a bug with Facebook, we'll report it in the next day or so after we hear back from you on the above.
Thanks,
Alex
Strange log out/log in problem
