Well I don't know about the shutting down during boot up but what I have noticed is that when you install/upgrade to W10, your power settings are automatically switched to some kind of laptop/tablet power saving setting even if you have a PC.
My system was doing the same thing for a while and I was a bit confused. I checked the power settings and the system was set to go into sleep mode after a period of time. Based on how my PC is built, when it goes into sleep mode, there are no external lights flashing and it appears to be completely shut down. When I hit the power button, everything powers up like it was off and then after a few seconds I was at the login screen.
The only thing that made me think this was different from my normal shut down was that it loaded a little bit faster. That made me think to check the power saving features and there it was. The thing is, the Sleep Mode in W10 doesn't seem like the Sleep Mode in previous Windows OS's, but is more like Hibernate Mode.
As soon as I turned off the sleep mode, I never had these mysterious shut downs.
It may not be the problem some of you are having but it sounds like what I was going through.