It is possible that the problem cropped up when we updated FileMaker Server from the original 14.0.1 installation to 184.108.40.206
The times that the update was applied and when users reported FMGo 14 stopped working could be sequential.
After checking with the iPad and iPhone users, it still seems plausible that the problem cropped up after, and not before, the update to Server 220.127.116.11. I guess I will cross-post this in the Product: Server category.
It seems that if FMGo 14 is exposed to Server 14.0.1 and then Server is updated to 18.104.22.168 then FMGo 14 can't see the updated server.
In all instances here, DELETING FMGo 14 off of the iDevice and redownloading and installing it fresh solves the problem.
This doesn't address the cause, but fixes the trouble.