Tadas,
I'm not sure what's happening, but if I try to login with Joomla credentials, I constantly get an "Invalid Token" message. It seems like that page is completely cached and simply not loading anything else when viewed except the 'not logged in' page.
Also, if I click the Login With Facebook button 4 times in a row on the home page, and then click into your site somewhere, I see the "Facebook profile successfully connected." message 4 times. So, it seems the login is definitely happening, but that page isn't reflecting it.
I don't think this is a JFBConnect problem, but something with how that page is being generated.
Hope that helps,
Alex