Well, after starting this thread, having my Hopper 3 crash, having to have it replaced, losing a lot of my recordings, I will tempt fate again.
Since I got the new Hopper 3 my EHD has showed up every morning.
So I have to assume all of my problems were related to my Hopper 3 and also then that others who have the same problem I did have the bad luck to have Hopper 3s that create this issue. No idea why one Hopper 3 will lose the EHD almost every morning (not every morning, but almost every morning - and it was taking more and more reboots to get it to show back up) and another will not, with all other hardware staying the same, but that's my experience after a couple of weeks or so.