Thanks to everyone for their advice. I finally solved my problem, and I'm a bit surprised nobody else thought of it first! It turns out it wasn't the wiper blades (or the windshield, for that matter) that was the problem. After much head scratching, and a certain amount of swearing, I had an epiphany! I lifted the wiper arms up, turned on the wiper, and low and behold they still squeaked! The problem turned out to be the linkage between the wipers. I got that lubricated and it quieted down. I will, however, have to replace the linkage, but that's for another day!