It's strange you're running into issues with LinkedIn authentication. I just tried on your site and it seemed to work the first time by bringing me to the profile selection page. On selecting a type "... professional", my edit profile area had my name, a username and email pre-filled. I didn't follow through to actually create an account, but can if you'd like me to test it out.
With that said, I'd double check that things fail in other browsers. Depending on the type of browser your using, and the add-ons you have enabled, social networking features may be intentionally blocked. There's a lot of 'privacy' extensions that stop almost all tracking, which also stop most social network functionality.
I hope that helps get you started, but if you have any other questions, please let me know.
Thanks,
Alex