Drew,
Obviously, that's not right. Are you using the new 3.2 version of JFBConnect, or are you on 3.1? In 3.1, the issue you describe above would likely be caused by an incorrect ordering of the JFBConnect User plugin in relation to the Joomla User plugin. (JFBConnect needs to have a higher number).
In 3.2, we haven't seen that issue, and may need to investigate more. Fortunately, there's no ordering requirement on the User plugins anymore, so that wouldn't be it. My best guess would be that you have a User plugin that's trying to redirect the user (or log them in). When that happens, the remainder of our login process doesn't continue, which could prevent profiles from importing.
Please check the above, and if that doesn't help narrow it down, let us know, and we'll come up with other things to check, or ask for credentials to check out your config ourselves.
Thanks!