Well that's definitely not good, or right.
Can you check to make sure you installed the Joomla 1.6 version of JFBConnect and any associated modules? If so, then check the following:
* If you have any other Facebook extensions installed, unplublish them, or disable any Facebook integration in other extensions (JomSocial, CB, jReviews, etc).
* If you have any SEF extensions enabled, try disabling them.
* Narrow down the specific plugin or module that's causing the error. Try enabling just the authentication, system, and user plugins. If you have any profile plugins installed, disable them. Do the same for the Login module, try disabling at first to see if you still have the issue.
As for the 404 error itself, is it a 404 within Joomla, or is it an Apache error saying the page couldn't be loaded, or some other type of error? Knowing what's actually happening will help us come up with a solution.
Finally, if you get to the point with the error again, please post (or private message) us a link to the page so we can see the output.
Thanks,
Alex