David,
Joomsef definitely looks to be destroying our URLs. I don't know Joomsef very well, but usually, 3rd party SEF extensions have options to 'skip' certain components. If you can find that setting, please have it skip JFBConnect. We have very specific URL requirements for communicating on your site and with Facebook, and if they get changed too much, bad things happen.
If you can't find the setting that makes it work, just let us know, and we can do a little more research.
Thanks,
Alex