App Rejected 3 Times, clueless how to fix issue so an App of mine has been rejected 3 times.  Each being the same reason:
Guideline 2.2: Apps that exhibit bugs will be rejected

More specifically the App Reviewer(s) cannot log into the Application Server, which is needed to proceed.

The problem here is that I can not for the life of me figure out why they can't login.  The App has been tested thoroughly around the World from various testers, in the States (N.Y and California) as well as in India and Australia.

Everyone can login but the Apple Review team just can't.  I asked them to test on actual physical devices for the last submission since I believe they are using their own test suite when reviewing Apps.  Not sure if they did though.

How should I proceed with this?  I'm out of ideas.
If Apple reject your app due to a technical problem, they usually provide a screenshot. Check the bottom of the message in iTunes Connect to see if there is an attachment.

You could try some severe logging: in the app, log exactly what is happening, all the inputs, all the errors, and have it report to your server. Thus, after they reject the app, you can go through the logs and see what is happening.