I cleaned the CPU fan as much as I could with a can of compressed air. I wanted to oil the bearing, but the bearing isn't exposed on either side of the fan, so sadly the "chirping" that initially made me want to clean the fan hasn't been remedied...
Now that you point it out, I don't believe it is as completely attached as it could be. I tried fiddling around with it and seating it better, but it's a pain in the ass to try and hold the fan in place, hold the backplate in place, and screw the screws with only two hands.
You shouldnt be needing to hold the backplate on.
The backplate attaches to the mounting brackets that hold the heatsink on the cpu. Then you attach the heatsink to the mounting brackets. When removing the heatsink, the backplate should not be coming off with it as its attached to the motherboard itself via the mounting brackets. Once they are installed, removing the heatsink on that mobo should be as simple as unclipping it from the mounting brackets. The brackets need to be screwed down properly to the backplate BEFORE installing the heatsink. Im not sure how you installed these AND the heatsink at the same time. Did you not watch a video on it before installing it?
Other than that, It sounds like the pc is overheating due to poor heatsink mounting or the plug for the fan on the heatsink is not plugged in all the way to the mobo causing the fan to not spin. Could have unplugged it while cleaning it (but you said the noise is still there so it must be spinning). Also, could be the fans "squeeking noise" indicating the fan is damaged and is not working at full potential causing high enough temps for the system to auto shut down.
Does the fan spin? I would remount the entire backplate to the mounting brackets and then mount the heatsink with new thermal paste.
Also, download HWMonitor here:
https://www.cpuid.com/softwares/hwmonitor.html
Install it. Open it. Scroll to your CPU. Under it, it should say Temperature. What is the temp of your pc when the crash is about to happen? Also, what is the temp of the pc as soon as you turn it on? Also, look at the spot where it says "Fans" under your Motherboard name, what RPM is the CPU fan spinning at? Must know this info to know if its a thermal issue.
Here is a video to aid you in reinstalling the backplate/mounting brackets and heatsink properly:
View: https://www.youtube.com/watch?v=yzWPwCp9ef