Matt,
When I load that page, if I go to the Javascript Error Console in my browser (Chrome), I'm seeing the following message:
Uncaught Error: You must specify a valid JavaScript API Domain as part of this key's configuration.
Check that the Javascript API Domain and Website URL in your LinkedIn Application is set to:
www.qualifize.com/
If those seem correct, then it may be a general communications issue. I just tried to login with LinkedIn on your site and it took about 45s or so for that error message to show. That may be an indication that there was simply a timeout with communication and nothing is really wrong with your Application configuration. In that case, can you check your System Information in Joomla and go to PHP and search for "cURL". Past whatever information is in that section, mainly the version number that you have installed.
Thanks,
Alex