Is a squeaking feeling in one breast normal when massaging pressure is applied? Does this mean the implant dropped?

When massaging implants they can sometimes give a speaking sensation on the chest wall.  This can occur with smooth implants early in the healing but usually goes away with time.  

You do not mention how recent your breast augmentation was. If you are in the immediate  post operative period, squeaking is associated with fluid around the implant. This is usually is self limited and resolves quickly.However, if you have had your implants for a while, you  should follow up with your surgeon.Yours,Jon M. Grazer, MD, MPH, FACS

You certainly should not hear any squeaking, but you might feel the implant moving.  I don't think this means the implant dropped.