Wow... I've never seen this issue before, but the problem isn't directly caused by JFBConnect. Here's what's happening:
* The JFBCSocialShare module is enabled and showing content from other networks
* You have some extension on your site that's loading the following file "/geeup/media/plg_system_externalfavicon/Mootools-External-Favicon.js"
* That Javascript is trying to load the favicon (the little icon in the tab of a browser for a site) for any URL that is loaded
* Twitter, for some reason, uses a favicon over http...
twitter.com/favicon.ico
That favicon is the only non-https content loaded and, again, it's not being called by anything JFBConnect is specifically doing.
I'm not sure what that Mootools-External-Favicon.js file is for, but I'd disable it.
I hope that helps, and keep us posted on what you find!
Thanks,
Alex