Login using JFBC Login box gives me 404 error

I'm not sure. It's likely not JFBConnect, but from the description, it's impossible to tell.

I'd recommend:
* Disabling SEF URLs
* Enable the "Debug System" option in Joomla -> Global Configuration
* Try to login again. In the URL bar, look for what option=com_XYZ is. com_XYZ is likely the extension that is not being found, but the error could be coming from a system plugin or something else that's missing what it needs

Hopefully that helps get you started. If not, I'd recommend disabling user and system plugins to narrow things down further.

Thanks,
Alex