Sagar,
I just created a subscription for you and you should be able to download JFBConnect from the My Account -> My Downloads area in the top right.
As for the authentication process, we use the standard JFactory::getApplication()->login('username', 'password') call to let Joomla handle the authentication. Our authentication plugin validates the user against a social network, and then Joomla calls the standard onUserLogin, onUserLogout, etc calls. We don't call them directly because we shouldn't need to. The login authentication code is in the /controllers/login.php file, though that calls plenty of other code based on the social network provider, if the user is new, etc.
Please feel free to get in touch here, or through our Contact Us form if you have any other questions. I did receive your Contact Us submission with the same post as above. It's simply easier to respond here, but we can easily talk through email or Skype if you'd prefer.
We'd love to hear about any issues you run into.
Thanks,
Alex