A squealing toilet annoys and can disrupt others' sleep when you get up in the night to use the bathroom. The squealing problem will require you to either swap out the ball cock in your toilet or at least replace the old gasket in the ball cock, which is the source of the squealing.
Ball Cock Setup
By taking off your toilet tank's cover, you will see if your toilet still has an old ball cock valve, which delivers the water from the water supply line into the toilet's tank. If you see a large float on the end of a metal arm that attaches to the other end of the valve, your toilet has an old ball cock. The ball cock's valve opens all the way when the float drops down to its lowest level when you first flush the toilet, but the valve closes little by little as the float rises as the tank refills. The partially open valve will emit a squealing sound.
Your toilet's ball cock uses a rubber gasket to seal off the valve, keeping water from flowing into the tank when the valve is in the closed position. Like other rubber parts in your toilet, the rubber gasket will wear out and become brittle over time. A worn gasket will cause the toilet to shudder and even squeal. Removing the old gasket and installing a new one may eliminate the squealing, although the ball cock will still make noises.
New Fill Valves
The fill valves that deliver the water from the water supply tube in modern toilets operate differently from the old ball cocks. Instead of the valve closing slowly, the valve stays all the way open until the float reads that the tank has filled up. Keeping the valve completely open avoids squealing or other noises coming from the valve, like in ball cock valves.
You do not need to be a plumbing pro to remove your old ball cock from the toilet and install a new fill valve. You must drain the water from the tank by flushing after you close the water valve. You will also need to use a sponge and bucket to sop up the rest of the water in the tank. You must then remove the nut that holds the water supply tube to the bottom of the tank and then the bolt that holds the ball cock to the bottom of the tank. You may need to hold the bottom of the ball cock inside the tank in place using a wrench. You then slide the new fill valve in place, and replace the bolts to hold the fill valve in place and attach the water supply line.