When I tried to login, I'm brought to the Login/Register page of JFBConnect, but see a "Facebook API Error: Invalid OAuth 2.0 Access Token". When looking at the source of your page, the inclusion of the Facebook Javascript library is missing. Can you check to ensure that everything on the JFBConnect Admin Overview page is showing there are no errors. Specifically, check that the JFBConnect System Plugin is enabled and the correct version.
If that's good, let us know, because we'll have to investigate further as to what's happening, but without the library being included, things will definitely go wrong.
If you're using a site optimizer, which is meant to combine Javascript files and other things like that, please try disabling it. I don't think that's the case from looking at your code, but want to throw it out there.
Thanks!