If you're getting the "Connection Reset" error, it likely means that your server is not able to make outgoing requests to Facebook. That could be due to a firewall on the server blocking those connections or a server configuration error, like invalid DNS entries.
I'd recommend trying to go to the Configuration -> Facebook area and setting the "Disable SSL Certificate Validation" option to "Yes". That could fix Autotune, but likely won't.
The next steps would be to try and either contact your host and see if they can determine if there are any connectivity issues to the following URLs from your site:
graph.facebook.com/
Another option is to go to the
Facebook Developer Area
and setup the options manually per the recommendations shown in Autotune. Then, you can test Facebook authentication on your site to see if there are any issues.
Finally, are you seeing any issues with authentication using the other providers (Twitter, Google+, or LinkedIn)? That will help understand the issue further as well.
Thanks,
Alex