First, nice looking site and good idea
As to your issue, I just tried to register using Facebook and, sure enough, it did a loop. I think I was then blocked by a firewall or something as I can't load the site again. What I can suggest is:
* If you're using a 3rd party SEF extension, configure it to 'skip' altering JFBConnect URLs. Alternatively, try disabling it altogether (temporarily) to see if that fixes the issue.
* If you have a caching extension or have browser caching enabled, please disable that to test. When I clicked the Facebook button, the page redirected very, very quickly, which makes me think your server returned a response before it even tried to contact Facebook.. which doesn't seem straight.
* Disable whatever firewall / security addon you're using that seems to have blocked me. It may be that the request JFBConnect is creating is being seen as a malicious attack.
* Unblock my IP (73.27.166.192) so I can test further
I hope that helps get you started, but let us know how things go.
Thanks,
Alex