(To JesseKnows: The squeal does not come from the display side, it's something around the Northbridge.)
I tried an automotive stethoscope. It's pretty hard to locate the sound with or without it. When I press the end onto components, they are transmitting the sound, since they are all attached to the same board. The Northbridge is louder than the fan housing above the CPU and also louder than the NVDIA chip, but it's about the same loudness as a number of small ICs nearby, and the round component (pot/variable cap/?) near the disk connector.
Two interesting data points:
1. When I press the display toggle button (next to the power slide switch), it generates a version of the squeal, even when the computer is off. It has always caused a red light to glow, but it now causes the squeal, which still seems to be coming from down near the Northbridge chip.
2. When the computer is off and I add power from the transformer, I get a squeal about 40 seconds later.
I'm having second thoughts about buying a new motherboard. I don't know what role the transformer is playing in all of this, and if other components are messed up, a new motherboard will only be the beginning of the costs. It still boots all the way up on the Ubuntu hard drive, but the squeal makes it unacceptable.
I'm going to look at alternatives again, and may part this unit out, or put it up for adoption. It's a shame, it's in such good condition other than the squeal... More news when it happens...