I just tested using Safari (on a Mac) and tried logging in using Facebook. I wasn't logged in or brought to the registration page, like you mention. I'm also not sure why that would be happening. Here are the steps I'd suggest to try and hopefully get you going:
* In the JFBConnect -> Configuration -> Facebook area, change the "Show 'Login Credentials' in popup" setting to "Yes".
* In the JFBConnect -> Configuration area, set "Enable Debug Mode" to "Yes"
Test with each of those and see if a) you can start logging in or b) a better error message is displayed on the page which may explain why JFBConnect isn't getting the user's credentials properly.
Not sure if that will get you going, but it should narrow things down either way.
Thanks,
Alex