I found some new information.
https://www.bleepingcomputer.com/news/software/avid-users-are-suddenly-finding-that-their-macs-won-t-boot/
I don't have any Avid video products installed, but it seems the problem is the iLok HW /SW dongle and it will destroy your Mac if you are using OS X 10.12 or lower. And it has been used apparently with many audio/video software packages, and while I don't do video, I do have some Audio plugins installed which probably are causing this (no, I don't have a USB dongle so this a software problem). And really, all my Macs currently using Sierra 10.12 or lower Yosemite 10.10.
The good news is that I reinstalled the Yosemite to my old MacMini through Internet (just do a Restart with command + R pressed down and follow the instructions) and the reinstall was successfull, indeed. I did not lose any data and the computer works.
The bad news is that if I will connect the MacMini to the internet, the problem will most probably occur again.
So hopefully iLok will find a solution with Apple.
Apple, please hear us: this is an untolerable situation!