Reza,
Looking at the HTML on your page, in the head section is the following line, which is including the Facebook Javascript library:
script src="http://connect.facebook.net/en_US/all.js#xfbml=1" type="text/javascript"
JFBConnect also includes this library, and does some other things with it as well. If it's included twice, you'll see the sporadic issues that you're noting above. Additionally, it will slow down the overall page load time. Additionally, at the bottom of the page is a bunch more Facebook stuff that doesn't seem to be implemented properly, specifically, look for FB.Init. The first one (toward the top) is JFBConnect. The bottom stuff is missing an AppID (required), and is definitely the problem.
Whatever other Facebook extension your using, please try disabling and I think your intermittent login issues will be solved.
Good luck,
Alex