Cars do the same thing, the squeak comes from the pads vibrating in the carriers (some manufacturers call these torque members). One way to stop it is to remove the pads and apply some disk brake grease to the very ends of the pads where they would contact the carriers. Also put a little on the back on the pads where the pistons contact the pads. This helps a little with the vibration but what it really does is lubricate the pads from any surface that they are mounted in. Now, when the pads vibrate they no longer rub against a dry mounting surface and it kills the squeak.
You don't want to slather this grease all over so it gets on the pad surface or the rotors. You can get disk brake grease from any auto parts store, it is usually high temp copper or mollybendium, sometimes its not cheap.
B&G also makes a liquid you can paint on the surface of the pads and it works like a charm The trouble is B&G only sells to shops, so Harry Homeowner would have to find a shop that would let him use some...
Because my well water is incredibly hard, I can't wash by bike at home. On the other hand it hardly ever rains here and the bike stays pretty clean so usually I just clean the windshield and use some windex on the rest