TestFlight is Currently Unavailable Message
Another possible reason for this message:
You accidentally removed the affected user from the TestFlight external testers group.
Adding him again would fix the issue.
I had this same error.
Discovered that the company certificate was expired, and therefor all provisioning profiles as well.
1. Renew the company certificate (download, install on your computer)
2. Renew the provisioning profiles (edit, select new valid certificate, generate, download and install in xcode)
3. Repost the app to iTunes connect
problem solved :)
If you're new to Xcode, be sure to note the difference between a CERTIFICATE (first large red arrow below) and then a PROVISIONING-PROFILE (second large red arrow below). Basically provisioning-profiles are made using a certificate. So first make a new certificate, and then set a provisioning profile to use that certificate.
First, certificate...
Then, provisioning profile...
In the second image, click on the "Distribution" provisioning profile (the one with the purple arrow.) Then click to edit it, and you will then see the third image below. Note that somewhat annoyingly, the DATE is the only way really to distinguish your corporate certificates, when, you have more than one. (Ultimately you can, generally, just throw away the OLDER one, but it's clearer if you leave it so that you can clearly see how you select one or the other, as in this third image.)
Editing a provisioning profile...
A minor point is that you'll have to make a new build (so, increase the build number in Xcode, then archive, and then send up that new one) because you'll get a "redundant build" warning if you fix the cert/profile (i.e., as described in this answer) and re-submit.