Dryer Isn't Level
Double-check to make sure your dryer is installed on a level surface. Place one hand on each side of the dryer and attempt to move it back and forth. If the dryer moves, then it's not level. When a dryer isn't on a flat surface, it can sway when it tumbles clothes. The movement puts pressure on various parts inside the dryer and can cause them the screech, rattle and clang. Modify each of the four adjustable feet at the bottom of the dryer until it's level.
Faulty Drum Bearings
Refer to your dryer's user guide to confirm whether your model has a rear drum bearing or rollers. On certain Maytag dryers, the drum is held in place with aid from a center spindle. The spindle is usually either a shaft within a sleeve or a ball-and-socket type of support device located at the back of the drum. If the bearing deteriorates, it can cause the drum to produce a screech or squeaking sound from connecting with the cabinet. Swap out the component to stop the noise. You can't fix a bearing once it has worn.
If your dryer doesn't have a drum bearing, then it has rollers. Rollers are small plastic or metal wheels on the outside of the drum that allow it to rotate evenly. They're apt to screech and squeal due to excessive wear. Unfortunately, you can't quiet worn rollers with lube; they must be replaced.
A Maytag dryer has a tensioner, or idler pulley, that applies tension to the drive belt while it rotates. It permits the drum to tumble consistently in a circular rotation. The part consists of a wheel, spring and bracket. If the wheel breaks, it's unable to maintain proper tension on the drive belt, and consequently the strained action causes a screeching sound. Locate the tensioner with assistance from your dryer manual, if needed. When it's in view, try to turn the wheel. If the movement feels forced and it's accompanied by a high-pitch sound, replace the entire component. In some cases, a faulty tensioner might also damage the drive belt. If the belt has cracks or fibers showing through, it needs to be replaced.