1064 - You have an error in your SQL syntax; check the manual that cor

Jan,
I just clicked the Facebook button and was successfully registered and logged in, as expected.

Can you provide the full MySQL error message that you're getting? That will help us understand which query is causing the issue. The only things I can guess are:
1) You have the User - Contact Creator plugin enabled and are creating/deleting users. That plugin has a bug where if you try to create a new contact with the same email address of a user that you've deleted, it will throw an error.
2) Some other testing issue where you're creating and deleting users and subsequent registrations throws the error.

Let us know if either of the above sounds right as well as the query error message.

Thanks,
Alex

The most common cause of not being able to log off is usually caching. Could you temporarily disable any caching on your site - Global Configuration Setting, Joomla's System Cache plugin, and any other caching extensions/plugins you may have, such as JotCache. Then try again and see if there's any improvement.

-Melissa

Glad that could help point you in the right direction. Let us know if you run into any other odd behavior.

-Melissa