The usual troubleshooting methods, forcing the phone to restart, etc., did nothing. The Apple Health app is built in to the iOS, so there is no way to delete it and reload it.
As far as I could find, Apple and the support blogs never mentioned this problem.
Faced with a phone not operating properly, the choices are to take it to an Apple Store (and probably wait a week for it to be sent back, and pay a fee), or to try a Reset to factory settings. That means that all the customization, downloaded apps, etc., are wiped out. Fortunately, I had backups on iCloud, and a months-old password protected backup to my PC (which preserves the passwords in the apps, etc.) So, resetting to factory, then Restoring a backup wasn't too bad.
But then came the bigger problem: iTunes wouldn't recognize my iPhone on either of my Windows 10 desktop PCs. The phone interface didn't pop up when connected by cable, and all of the menu choices under File_Devices were grayed out. So, there was no way to Reset to factory settings. Restarting the PCs and the iPhone did nothing.
Through some searching, I knew that there should be an Apple driver, Apple Mobile Device USB Driver, shown in the Windows Device Manager, under Universal Serial Bus controllers category, but there wasn't! More searching indicated that the way to fix that problem was to re-install iTunes. So, more time and effort doing that, but the result was the same -- no driver, no iPhone device in iTunes.
Hence the Catch-22.
I contacted Apple Support, both by Chat, and by calling on the phone (and giving up after half an hour on hold). I got nothing that helped, except the instruction to Reset the phone using iTunes, and to re-install iTunes, which wasn't possible or did no good, on either PC.
After much Google searching, I finally hit on one article that solved the connection to iTunes; once that was done, I could go through the the procedure and hope that the Health app would work (which it did, thankfully).
So here's the procedure for getting the Apple driver into Windows 10:
I am so very grateful to Mitch Bartlett for posting this fix! This is an eleven-step process, so I won't copy it here. It involves manually finding the driver and installing it to Windows 10 for the Apple device. Once that's done, iTunes sees the device, and the rest of the Reset and Restore can be done.
iTunes: Fix iPhone or iPod Not Detected in Windows 10I'm still finding apps that I have to re-enter the ID and password, that weren't in my previous encrypted backup. And I'm backing up my phone to the PC more often now, not relying on the iCloud automatic backup.
BTW, iOS 9.3.1 came out not long after this problem, but nothing was mentioned about fixing the Apple Health app, so I don't know if that would have done anything for my problem. Never saw a reference to my problem from the iOS 9.3 update. Also, fyi, I had been running the 9.3 Beta on two iPads, for all of the nine or so updates, without major problems, so I had felt fairly confident in doing the update from 9.2 on the phone. Never again will I update on the first day or days of release of an update!!!
Hope this post helps some other folks.