What is causing my gas water heater to squeal?
My guess is that something in the gas supply line of the heater is whistling (a vibration is being induced from the flow of gas). This could indicate a blockage or deformation of one or more of the gas jets. It's unlikely to be any problem with the exhaust flue or with the water in the tank, as you say.

The other thing it might be, depending on the tank's proximity to your gas meter, is that the gas reg on the meter (which steps down pressure from the gas main to a level suitable for your home's appliances) is starting to fail, and the heater, being closest to the gas reg, is picking up a vibration through the gas line that is induced by the failing reg. This is less likely especially if no other appliance makes the slightest noise, and the reg itself isn't making any noise when gas is flowing.

As far as fixing it, this is something I'd leave to a pro, but that's just because I know enough about gas to know I can seriously screw it up. The fix would likely be to replace any obviously crushed or deteriorated gas pipes and fittings in the HWH installation, and possibly disassemble the line back to the wall to make sure there aren't any nicks or dents in the pipe ends from where the installer dropped and damaged one, then installed it anyway.