I'm not exactly sure what's causing the conflict, but you have some Javascript errors on the page which are preventing our Javascript from executing the actual login process. When the page loads, the following comes up:
Uncaught TypeError: Object has no method 'addEvent' -- modal.js
Uncaught TypeError: Object [object Window] has no method 'getStyle' -- mootools-core.js
Again, I don't know what extensions or what in general is causing those issues, but they aren't related to JFBConnect. Once the Javascript errors occur though, it can prevent future Javascript from executing, which is the problem you're running into.
What I'd try is:
* First, test with a different template to see if the issue goes away.
* Then, try disabling any Javascript heavy extensions you may have
Once you narrow things down, we can help further or hopefully you can contact the developer of whatever extensions are causing the Javascript issues.
Hope that helps,
Alex