Okay, I got High Sierra to install. The problem was with the EFI partition, which was the reason for the firmware error message. This is what I did:
I booted into the recovery disk on an external drive. I **don't** honestly know if this is necessary. It may be possible to boot into the recovery disk on the main MacHD drive but I did not try it.
In the recovery area, I opened Terminal from the Utilities menu. I then entered diskutil list to see the address of my hard disk. It can also be seen in the Disk Utility; it's probably disk0. At any rate, I wanted to see the partitions on the disk which is not possible in the Disk utility. If your disk is not disk0, then you need to use the proper disk number in the following commands, if you choose to try this method.
Then, in Terminal I entered diskutil verifydisk disk0 to check the disk. Sure enough it said that there was a problem.
I then ran diskutil repairdisk disk0 and got a warning that it would probably erase a certain partition and may cause a problem booting up. The partition it referenced was the EFI partition, so I said yes. It repaired the disk, I rebooted and then (finally!!!) installed High Sierra without a problem.
I could not do this repair from the Disk Utility. It had to be done in Terminal.
I also understand that a reinstall of the current OS on your Mac will fix the EFI partition. This can be done in the Recovery Disk menu, Reinstall Mac OS. Don't bother looking for Sierra in the App Store. It's not there. But it is available by download from the Recovery Disk. I did not try this method because I didn't have the patience for one more MacOS download and install. But you're welcome to try it, of course, if that's easier for you. You would reinstall Sierra and then upgrade to High Sierra if you go this route.
I hope this helps those of you who are still having problems. Regards.