Did you know that most chewing gum is a petroleum product? The original source of gum base was from tree sap but in our modern age this has been replaced with synthetic polymers. Chewing gum is mostly a delivery system for sugar but there are some health applications such as tobacco abatement and teeth protection. You might want to consider some of these healthy alternatives:
Licorice root, sunflower seeds and nuts are good if you just want to chew on something. If bad breath is a concern then parsley and (surprise!) water are great options.