Can a broken mouth guard cause jaw pain?

Yes: A broken mouth guard has the potential of causing jaw pain. Although a mouth guard is most often used to protect teeth, it also affects the biomechanics of the jaw muscles as well as the jaw joint.

If the original: Unbroken mouthguard was causing relief, but now the broken one is causing pain, then it would be reasonable to assume that the broken one is causing the discomfort. I would recommend being seen by your dentist to determine the cause. I am assuming the mouthguard is in fact a "nightguard" and not an atheteltc "mouthguard". In the interim stop wearing the broken mouthguard.

Sure: Did the mouthguard create relief for you before breaking? If it did, then the break may have altered the fit. Either the mg is not fitting correctly or the teeth are now not occluding on the mg the way it did before. Also, if you are not wearing at all, then the muscles and the jaw joints may be reacting to the change in the bite. See a dentist or tmj/tmd specialist asap.

Mouth guard?: Many patients use the terms "mouth guard", "night guard" and "splint" interchangeably. Another term is orthotic. I assume you mean a device to treat jaw joint problems (TMJ) or bruxism (clench/grind of teeth). If so, yes. The device must fit accurately for success. If the "mouth guard" was made by a dentist, return for repair/remake.