Use something else to release stress / energy / stay focussed. Gum is unnecessary and hard to remove. Most of the arguments for explain how it can help students concentrate. Why not use a stress ball, or a hand grip exerciser?
As for keeping them quiet? They still speak with their gum in, only with less eloquence and a white bit of plastic rolling around their mouths. Plus the teachers will want them to occasionally speak up and answer/ ask questions.