I finally found the right sized and color "round to-it" yesterday and updated my iMac to
macOS 10.12 Sierra. Unlike the mini, funning WC, it shuts down each night and restarts the next morning requiring a password. Today, when I finally got to sit down in front of the iMac, I found a huge list of "Disk ejected improperly, don't do that!" notifications. The login prompt was still there, of course. I then learned the obvious; you can't remove Notifications unless you are logged in. The good thing is that one need not click/Close every single Notification. Apparently, maybe since they were all the same message, a couple of Close clicks got rid of all of them.
My next, un-exciting task was to figure out which of the two external drives was the problem and why it/they were ejecting themselves. After several swaps with the four possible drives, switching cables, etc. I finally remembered the "sleeping" computer problems in this thread.
Sure enough, upon opening the Energy Saver System Pref, there was the "Put computer to sleep" setting at 10 minutes.
I changed that to "Never" and have had no more erroneous disk ejects, even though the Display has gone to sleep several times.
What is confusing to me (along with much of "life") is that I still have "Put hard disks to sleep when possible" checked! Apparently it is the computer going to sleep rather than the disks that causes the "Disk ejected improperly" Notification. Seems backwards/counter-intuitive to me...