Not out of the box, but it would honestly take about 3 lines of code to hide all comment boxes completely if the user isn't already logged into the site. We can gladly help you with that, if interested.
We would *not* be able to help you to show the box only to a user that is in a specific Facebook Group though. That's much more difficult, and I'm not sure how long that would take, or even if it's really possible.
Hope that helps,
Alex