I chew gum a lot at work. Is it true that your jaw gets bigger when you chew gum?

Jaw muscle growth. Yes this is possible but highly unlikely- unless you are chewing on taffy all day. If there is enough activity in the muscles of mastication there is a chance of muscle development.
TMJ. At age 52, no regardless one is chewing gum habitually or not. It can adversely affect the temporomandibular joint though.
No. The jaw is a bone, and like other bones it stops growing once a person reaches adulthood. The jaw should not get bigger regardless of what one chews.
Muscles. Your jaw muscles are being overworked by excessive gum chewing. The masseter and temporalis (Google it) will get larger. Square jaw appearance.