Can a dentist break an implant stud when removing the crown?

Not likely: Not sure what you mean by stud. I don't think it is possible to break and abutment. The second part of the implant that the crown is cemented to. However, the abutment is held in by a screw. Screws do come loose and have been known to break.

Very difficult: In very rare cases my be , but normally a very easy task to do, depending on the situation in difficult cases we'd rather compromise the crown ( by cutting it ) not the implant.