It's called a Veggie burger because it is typically a patty made of ground soy and other vegetable matter. xD
As for the chicken question, we read the menu to see the description of the sandwich. From there we're informed what type of chicken, bread, and toppings it has.
As far as what a burger is called whether it's inside or outside of the bun, I think it's time you met the hamburger steak.
Welcome to cultural differences 101.

