Does it ever hurt the same way when you don't do that? It's certainly possible i woud think that the mechanical actions of swallowing could pull on the cervical vertebrae. On the other hand, acid reflux and other similar ailments can cause lots of weird pains in places you wouldn't think would hurt. It might be a lot easier (and cheaper) to investigate that idea before assuming it's something wrong with your back - especially if you haven't had back/neck problems before.
Let us know how things are going,
David