Sorry for the delay. The problem looks to be due to your multiple jQuery inclusions on that page, including a very old v1.6.4 at the bottom of the page in this block:
<br></div></form><!-- END: Modules Anywhere --><!-- START: Modules Anywhere -->
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.6.4/jquery.min.js" type="text/javascript"></script>
<script>
$(function () {
$("#btnAddToFavourites").click( function () {
$.ajax({The jQuery libraries are being loaded about 4 different times, including that last one. I'd recommend removing as many inclusions, especially the 1.6.4 version, as you can as you really should only need to load jQuery once. The newest version on your page looks to be jQuery 1.8.3 and that should be backward compatible with all the other versions.
Finally, one last change you can make that should also fix the issue is simply change the "Logout of Joomla only" setting to 'Yes'. That will bypass the functionality that's preventing the logout but will only log the user out of Joomla and not Facebook.com
I hope that helps,
Alex