It looks like the div for the JFBCSocialShare belongs to your main container, so it is only clickable when you scroll down to this element.
Could you try adding the following script to the module default.php file? It will attempt to put the module right before the closing body tag so it's outside of every other div. We do this in SCLogin, but haven't had anyone report this as a problem for this module.
I put the module in an "aside" position that is, yes, below the top area where the nav lives. I suspect had I put the module in the nav, that might do it for the top, but I wonder what happens when I scroll past. Forcing it into the body seems the best solution.
Will you update the module as such, or should I expect to do this in an override?