This may or may not be helpful, but this reminds me of a similar experience I had...twice!
My old computer would regularly freeze (specially when playing games) and reboot...and sometimes also give blue screens at random moments. I could never really figure out what it was, so I started to experiment. After removing some RAM, computer hardly ever froze, but I still never really found out what it was.
Fast forward a bit and I bought a new computer that I also built myself. Thinking that everything was new and fresh I was confident that at least this time there would be no problems. I installed Windows fine and when I started copying data from my old HDDs to the new ones, I got random blue screen. I rebooted then tried again...got another random blue screen...
Long story short, in my case at least the cause was Memory settings in the BIOS, so I had to enter the BIOS and change the voltage and enter the CL numbers that you'll find on the Stickers on the memory units. After doing this, my computer has been super-stable since 2007!
TLDR - You might want to check your memory settings in your BIOS setup, to make sure they match the numbers on the sticker on the memory units themselves. And even more importantly make sure they are running at the correct voltage, which was the basis of my blue screen issues.
And like I said, this might not be a solution at all, but when I read "new computer" and "blue screen" it just reminded me of my own experience.
Also, after going back to the old computer and doing the same with it, it was also working flawlessly.