Hanna,
JFBConnect doesn't alter any Joomla files when it installs. The only way our extension does anything is if our plugins are enabled. If you're still confident that JFBConnect is the cause, I'd recommend uninstalling it entirely. All of our settings remain in the database so when you re-install you wouldn't have to re-configure everything. If JFBConnect isn't present on your site and the problem persists, it's definitely not a JFBConnect issue.
Beyond that, this thread started in October and you didn't seem to notice the Joomla login failures until mid-December. My guess, and what we've seen in the past, is that users start testing their authentication features on their site more when JFBConnect is installed. This simply brings to light issues that already existed before, or changes made after, JFBConnect was installed.
Thanks,
Alex