I just tried on your site to register and login with Twitter and I didn't run into any issues. The first image below is when registering and then the second is after logging out and logging back in:
So, the good news is that it's working for me. If you're seeing a blank page, that means there's a PHP error somewhere. What I would recommend is:
1) Going to the Joomla global configuration area and setting "Error Reporting" to "Maximum". Then, try again and see if a better error message is shown instead of that blank page.
2) Try logging out of Twitter.com and back in. Then try to re-log in. Alternatively, try a completely different Twitter account, just in case.
Let me know if either helps narrow things down, and we'll gladly help however we can.
Thanks,
Alex