We've had non-stop rain here for a few hours in Gaston/Lexington/Irmo, SC, so I had to look this up after my irritating 40 minute ride, each way, dropping the kid off at his dad's for the weekend. Figured I'd share the tips :)
How to stop windshield wipers from squeaking
Clean the windshield glass thoroughly. Try one of these products (but not all at once):
Baking soda (sprinkled on a dry or damp paper towel).
Replace your windshield wiper blades regularly. Wiper blades are not terribly expensive, so you may prefer to get good ones. Replace them at least once per year, before the rainy season.
Refill your windshield wiper fluid.
Check to see if your blades or the wiper assemblies under them are too loose or too tight. This could be causing tension between the glass and wiper.
Apply a product like RainX to your windshield. RainX isn't a glass cleaner but a water repellent that could help to fill in the tiny irregularities that cause your wipers to chatter.
Make sure that your windshield glass is really clean without waxes and oils. This could make the glass very slippery and cause problems.
When you have a free day, try various methods in case one does not work on your particular car.
Don't follow other cars too closely. The road splash contains rubber compounds that build up on the glass that can cause chattering. The film on your windshield builds up quicker when it first starts raining.
If your wipers only chatter when the windshield is not wet enough, clean the windshield thoroughly and try not to worry too much. Lots of wipers skip when the windshield isn't quite wet enough.
Never wax your windshield as this could make the glass and blades very slippery and prevent you from seeing in harsh weather conditions.
When replacing your blades you may have to go through much trial and error because this depends on the type of car you own and how a particular type of wiper blades fits.
After finding a perfect new set of windshield wiper blades be sure to keep notes on the brand name and type so you don't have to go through this ever again.