Dirco,
I just tested with Chrome on Windows. I created an account, was automatically logged in, logged out, and then back in again. I didn't run into any issues.
Can you try from a different browser and see if that helps? If not, temporarily enable the "Show Facebook API Errors on Front-End" setting (Configuration -> Facebook API tab) and let us know if you see any errors during logging in.
Let us know if you get it going. If not, we'll do what we can to help!
Thanks,
Alex