2001 Toyota Corolla squeeking. Getting louder

Since your car is front wheel drive, there is no rear differential. Someone is pulling your leg, lying, or you have misquoted them.

If from the rear, the squeak could be any of several suspension bushings, or the trunk lid rubbing against the body, or doors rubbing against gaskets. The latter two are more likely if the car has been hit in the rear, even if only slightly.

Your really haven't provided enough information to make any suggestions about replacing the transmission. Common danger signs there are if the trans fluid (which should be a medium red, and clear like cherry soda), is brown, contains particles, or smells burnt. More than that requires much more information to make an assessment.