iPhone 12 randomly shuts down

You may have a hardware problem

You already trief a forced restart.

If that does not help the  next procedure would be to backup then restore your device to factory, and setup the device without restoring a backup, to see if the issue is replicated. 

Then restore the backup. 

If you need to send the phone in for repair then backing up and restoring to factory would be needed anyway.

Here is a guide for those steps

Backup and restore (I recommend the computer method if you have one)

(1)How to back up your iPhone, iPad, and iPod touch – Apple Support

Remove SIM PIN if you have set one

(2) Restore your iPhone, iPad, or iPod to factory settings – Apple Support

Set up as new from the Hello Screen without restoring backup. Test if the problem still exists

If the problem goes away in the newly set up phone then suspect settings or apps, and in particular VPNs and so-called security apps which should never go near an iphone.

If the problem remains then arrange a repair including details of these tests.

https://support.apple.com/en-gb/iphone/repair/service

(3) When your phone comes back Restore your iPhone, iPad, or iPod touch from a backup

You have two phones and both of them shut down by themselves all the time?

If you really have not installed security apps and they both do it if restored to factory settings and set up manually without s backup restore then yes they both have a hardware problem.

That would be so peculiar that

1. the iOS has really has a flaw or
2. You are confused about what you have installed or
3. you didn't reset to factory to test it

I don't believe option 1.

I haven't done anything special to our phones, she has been complaining for a while and I've ignored her until this weekend when I remembered to update mine.

Unfortunately I don't have an Apple store where I live... Has there been a recall or update that I can add to my current phone? This has now happened three separate and very inconvenient times... and, unfortunately when I try to login for Apple support, I cannot due to the two factor authentication that goes to my Iphone (which is NOT working).

Any ideas?

"....If the issue only comes back after I do a restore that "could" point to software issue (?). I would then try to setup the phone again from scratch by setting up the individual MFA accounts again. ..."

Yes

"....when I try to login for Apple support, I cannot due to the two factor authentication that goes to my Iphone...."

Put the SIM into a spare phone to Receive 2FA codes then add a family member's phone number as trusted number in appleid.apple.com security section.

iPhone XR here, sames happens to me and my mom (iPhone 12 pro) since we updated. I guess we all have a hardware problem?

If two phones randomly shut down in one family then consider what you have done to both phones - VPN or security for example.

My reply that was recommended by Apple is the way to go.