I don't understand the teacher's reasoning. If your son is allowed to play quietly alongside the other kids in the classroom, why can't he eat his lunch quietly with the group as well? How is he supposed to become more comfortable talking to his classmates if he is not even allowed to be with them? And besides, why should he *ever have to talk if he doesn't want to? There are plenty of reserved adults in the world, and they are not barred from restaurants and dinner parties because they prefer to smile quietly at their tablemates' loud jokes or have a hushed conversation with just one close companion. Really, this is just wrong.
I think your DH is right, you should talk to the director. By the time the teacher deems your son "ready" to join the Lunch Bunch, he will have missed out on a month or more of bonding, and will find it even more difficult to integrate himself into the group. Being excluded is also likely to hurt his feelings, as you said, and make him self-conscious about his natural reserve. I hope you and your DH can make your case to the school on this one.