Personally I don't like the idea of chewing gum especially the sugary ones maybe the mints could be a better alternative but I don't vote for it.I prefer taking liquids,for now I'm taking more of water I don't need anything that will spike insulin.
I had a habit of that same thing before, chewing gums and eating candies whenever I feel hungry. And I actually learned here in this forum that doing that will not help me from my diet. I stop it but gradually stopping it since it was a hobby already. What I did is I drink water instead, lemon water rather. And fortunately I avoided doing it again.
In terms of calories, I’d definitely say that the calories in chewing gum are pretty negligible when fasting. However, there is another reason you shouldn’t chew gum while fasting.
Studies have shown that our body triggers an insulin response simply due to the experience of seeing food (like on a cooking show, for example), because it starts ramping up the hormones in preparation for a meal. Now, what do you think happens when you actually put something sweet in your mouth? You're convincing your body that it’s going to have something to digest, therefore triggering an insulin response, and therefore continually storing fat in your body.
For a pure fast, I would only go with things like water and tea, preferably just water.