I did a search and didn't see this but I could be wrong and if I am being redundant I am so sorry, it's not my intent.
 
Original purchase was in 02/2015, I had the card replaced in 02/2015 for this exact issue.  Whenever I play a game the entire system reboots, it does not matter what game or if it will reboot but when; some games reboot it immediate others take a half hour but it will eventually reboot.  There is an audible click before it reboots and there is coil whine.  I had this issue with both cards and apparently out of warranty since it goes from the original purchase date not the date of the replacement.
 
I am running:
i7-4790K
G.Skill Trident DDR3 16GB
ASUS Hero VII Maximus
EVGA 850w PSU
EVGA GeForce GTX 970 4GB
 
I know it's not my PC, I replaced pretty much everything thinking that EVGA couldn't possibly send me a card with the exact same issues.  So what I have done on my end for troubleshoot, I put my kids' GTX 970 ti in my machine - no reboots no matter what game (but looks terrible because it's a downgrade), put my husband's GTX 1080 ti in my machine - no reboot no matter what game, wanted to keep his card but that was a no go lol.  I put my card in their machines and in both cases the the reboots followed the GTX 970 so it is definitely, without a doubt this card because no one is running identical hardware and their machines work great until I put the 970 in, same for me when I take out the 970 my machine runs top notch.
 
We are trying to put in an RMA even though it's out of warranty given it is a manufacturer defect but I am not hopeful.  Does anyone know if there is a way to repair it or any other ideas?  If I could afford it I would be buying one of the new ones but 3 kids and life so need to be responsible, I am however desperate to get this fixed somehow.

You could try to bake the card back to health... https://www.overclockers.com/the-oven-trick-repairing-your-broken-video-card-with-an-oven/
 
Note: Bake the card with the gpu die facing upwards.

Bake it and hope for the best.

Hello,
 
Sorry to hear your having issues with this card? When you say the card is out of warranty have you contacted us here at EVGA Customer Support? Now as to your product are is it a  970 or a 980ti or another product?  I will send you a PM for further assistance so we can address this issue.

''that would cause the screen to freeze and then audio to get very loud and screechy?''
 
my ''last'' asus did this  and it ended up being the motherboard  , card seemed to work fine in my hand me down builds and even ran it in my buddies new gaming rig  fine  .
 
replaced the mobord and all was good .. [also did a lot of swapping struff and reinstalls of drivers and OS till all that was left was a motherboard , and that was the fix 
 
that comes dfown to being able to have ways to test hardware  out side of the system that has issues .   see we thought it qwasa something wrong with the card but  proved it was not  .
 
also try to do a custom install of just the cards driver and nothing else  [like don't install the NVidia hd audio driver and all unessary things added in the driver package  ] just the driver and se if that nidia hd audio driver is not crossing up that main audio driver like say if realtek's ] 
 
ya , that asus you could not get the speakers turned off and the computer  shut down fast enough  when that screeching started  . just kill it off at the psu switch or quickly unplug from the wll ...lol...
 
funny thing  was my buddies asus went bad after mine did in like a month   he replaced it with a giga pulled all the parts off the asus  fired it right up and been working fine ever since 
 
like I say '' my LAST 3 asus were my last asu's ''