You are connecting to your studio monitors via balanced cables, correct?
Using balanced cables will help minimize induced noise.
That said, I don't think it'll eliminate the noise being discussed here.
If the PS is making that much noise, it should be replaced.

I found some balanced cables, and switched to those.  The noises went away.  So far, at least.  It could be a coincidence, though, since I didn't listen to unbalanced cabled immediately before switching to the balanced ones.
But at least there is some progress, even if it turns out to be short-lived.

Okay, the speakers are quiet now, but the back of the computer is still squeaking frequently, not necessarily when I am moving the mouse.  I am currently downloading and ripping at the same time, and the squeaking is pretty constant.  Very annoying!
 
I am currently considering getting one of these:
 
a) EVGA SuperNOVA G2 750W ($209 NZD including shipping)
b) EVGA SuperNOVA G2 850W ($279 NZD including shipping)
 
Is that extra 100W worth the extra $70 ?
 
And are the SuperNOVA G2s good for a Sonar DAW?
 
 
 

To me it sounds like you are hearing two different noises and just associating them. Seems like a fan problem and a ground problem.

+1, the power supply fan is something that cannot be cleaned thoroughly unless taken apart, but dust will commonly make them wobble or even make contact full time. The noise from them is simple vibration and/or wear contact. If you want to take that apart, they are simple to clean/lubricate. This guy shows the entire disassembly - https://youtu.be/uqwgUtcXjPM You want to clean the entire fan portion, as well as lubricate the contact surface.

The power supply is about 6 years old.  Yesterday it (or something near it) was really noisy, squeaking like crazy most of the day.
 
I went ahead and ordered a new power supply (EVGA SuperNOVA G2 850W).  If that doesn't fix the problem, then maybe there's something wrong with the motherboard or the video card, neither of which I am eager to replace!

If the two noises are in perfect sync, the issue isn't a fan.
 
The noise of fan bearings going bad is (obviously) cyclical
Can sound like a repetitive "squeak" to more of a "grind" (especially smaller/higher RPM fans). 
If you've ever heard the sound of grinding brake pads... it's similar.   LOL
 
In either case, the OP would want to swap out the power-supply.  

Yeah, I sure hope a new PSU is the silver bullet!  The high-pitched shrieking is really annoying, since I am on this computer all day!