If the user logs in with Facebook, they can be automatically logged back in when they return to the site if they are already logged into Facebook.com. You can enable that option in the JFBConnect configuration settings for "Automatically login Facebook Users".
Beyond that, JFBConnect does not store cookies to force the user to stayed logged in when they use a social network. The problem with that is if they log out of the social network, the expectation is that they won't still be logged into your site if they stay away for a while.
One thing you may want to look at is your "Session Expiration" timeout. By default, this is set to 15 minutes, which means a user is logged out after 15 minutes of no activity. I'd recommend increasing that to something like 60 or more so that users aren't logged out unexpectedly when they have your site open in another tab.
I hope that helps,
Alex