Facebook login is definitely working on this site and others. Can you try with a different computer and/or browser? My only guess is that you have added an extension to your browser to enhance privacy, which is likely blocking accesses to social networks.
On your site, I can see the FB Like/Share and Twitter Share buttons.
However, I tried to login and the FB login popup showed:
Error
App Not Setup: The developers of this app have not set up this app properly for Facebook Login.
Can you re-run the Autotune configuration tool and make sure your FB App section is setup properly according to the recommendations?
I hope that helps, but if you still need more help, just let me know.
Thanks,
Alex