HAHAHA !!
I'm sorry, I don't mean to laugh, but I remember my daughter doing the same thing. We all used to laugh at her and the saying went through the family that my kid held food in her mouth until it liquified !
He will get over it, I promise.
In the mean time, do you have to give him chicken ? How about replacing it with something softer, like vienna sausages, spam, potted meats or balogna chunks ? If you really want to push the chicken, then process it with a little milk or baby formula. (It doesn't have to be completely pureed, just chopped finer. My guess is he will still do it though, until he gets some back teeth. (My daughter did.) It really is hard to "gum" no matter how tiny you cut it. When he gets more teeth, then try it again.
By-the-way, my daughter is 21 years old now, and honestly, I don't think she has held food in her cheeks for quite a few years ! LOL !
Hang in there ! If this is the worst problem he ever has, you are doing very well !
P.S.
For the veggies, just cook them longer until they get really soft, then cut them in small easy pieces for him to pick up. He sounds like a little guy that wants to feed himself, so it'll go easier if he can pick them up himself. The extra soft texture will make them melt in his mouth. You could even add a tiny pinch of sugar to the veggies to entice him.
Be sure you are giving him a vitamin suppliment, since you are having trouble balancing his diet.
:o)