That issue doesn't look to be a JFBConnect, or even a Joomla issue. Facebook simply can't get data from your site. Although a 200 response code is being returned (which is correct), Facebook isn't getting any data, and that's why they're throwing that message.
I've been looking, but couldn't find any reason that Facebook can't fetch your page contents. The best way to tell if this is a JFBConnect issue is to disable the JFBCSystem plugin and test again. That disables almost all of JFBConnect, including the Open Graph tags. It sounds like you already know that it isn't JFBConnect, as you mention other extensions have had this problem.
If Facebook still can't scan your page with that disabled, you'd need to investigate your access logs and see what is being returned when Facebook hits your site. Facebook has a User Agent string of "facebookexternalhit/1.0" or "facebookexternalhit/1.1", which should make finding the request easier.
To investigate the logs, you would likely need to contact your host. You should be able to check the response logs for the response code (it should be 200) and the length of the page which was returned. The length will be critical because Facebook is reporting that "No data could be retrieved from URL". My guess is that, on these requests, your server is returning an empty page, which is why Facebook is complaining.
Some servers have odd configurations that block or change the output when Facebook is detected. That's not the right behavior, and the best guess i have for what's happening on your site.
I hope that helps get you started. I've tested with multiple 3rd party tools to fetch your page, and they all come back with the correct body response. So, we'll need to narrow down if it's your server returning something incorrect. If not, then it means Facebook isn't accepting what you're returning.. which could mean your site or IP address has been blacklisted for some reason. That's more difficult to fix, as you'd need to contact Facebook.
I hope that helps,
Alex