Can I lubricate my drill?

Usually a squealing noise is caused by a bearing and not the gearbox. If you take it to bits and check the bearings on both ends of the motor hopefully one will feel stiff or rough to turn. Does the chuck wobble when moved by hand? If it does then that bearing could be the cause.

@rich53945 yes you can, right after you clean all the old goop off :-) Use something like this video to work on it. Let us know what you find out and if in any doubt, you can always post some well focused images of your driver with your Question. That way we can see what you see. Adding images to an existing question