There isn't a way for determining who has Like'd something on your site. Facebook intentionally prevents us from knowing this to prevent incentives, like you're trying to do, for clicking the Like button. The whole system is based on honest 'social' feedback, and they don't want it giveaways or freebies to be used to get user's to Like the site.
The best I can suggest is that you enable the Like Notifications (in the Social area of JFBConnect). Then, if a user is logged in, you'll get an email with their Joomla username as to who liked the page. There will be no way to know if they Like'd it when they weren't logged in, or if the later on un-Liked it.
Hope that helps explain, and good luck,
Alex