I've never heard of JFBConnect uninstalling itself, at all. There's nothing in our code, at all, that would do such a thing. Can you check that the directory for /components/com_jfbconnect and /administrator/components/com_jfbconnect is actually deleted, vs something in your database rolling back and unpublishing the plugins for JFBConnect? After you re-install, do you need to reconfigure JFBConnect, or are your settings still intact?
In either case, again, I'd be very confused as to what's happening. The screenshot is helpful in that there's a notice coming up that we should fix, but that's definitely not the cause of the problem.
Please let us know what you find on that, as there's nothing that should cause it.
As for the username, we have some instructions in the following post that should hopefully help. Please remember that usernames in Joomla must be unique, so you should use some method to create 'randomness' in the usernames (which is why we currently use the Facebook User ID) to ensure that if 2 "Tom Jones" sign up, there won't be a problem creating the 2nd tom_jones account:
www.sourcecoast.com/forums/topic?id=1981&p=1#p8885
Good luck, and definitely keep us posted.