I've tried to strip this problem down to its basics. I've completely powered down the Modem, the WiFi Router, and the computer.
Disconnected the computer completely. So there's no computer OS problems, windows problems, firewall problems, iTunes problems, etc.
Connected and powered up modem. then connected and fired up WiFi router, which has been reset to pure factory settings.
Fired up iPhones, each one separately. They find the wifi network, find HughesNet, get onto Internet fine---web pages, email, messaging.
I go to the App Store and search for apps. App store works great, I pick random free app. Try to download it. Up to that point everything's looking good... but the download never begins.
Got to Apple Store app to download music. Can browse, sort, see Already Purchased Music, everything. Pay for a song, transaction goes through. Go to Downloads... and song is stuck, never downloads.
(All of the above mimics what happens on my Desktop using iTunes. Apps, music; everything works EXCEPT the actual download... iTunes gives a more detailed error message than iPhones do; the iTunes Diagnostic tool under it's Help menu says 'Failure to Connect to iTunes Store".)
Drive away from my house towards town, and bingo, as soon as I get on the cell network, or pause at a McDonald's free WiFi, my apps are downloading, my songs download.
So I've stripped this computer out of the equation, and demonstrated that my iPhones are wanting to and capable of doing the connection to the iTunes Store... but NOT when the connection is HughesNet.
My only hope is that one other user a year or so ago said their Hughesnet tech 'switched my Hughesnet connection to another satellite network' or something, so maybe that's a thing?