Preheat oven to 350 degrees F (175 degrees C). Grease a 2-quart baking dish.
Place a layer of potatoes in the bottom of the prepared baking dish, followed by a layer of onion, a chicken breast, and a layer of squash. Repeat layers, season with salt and pepper, and dot the casserole with the butter pieces.
Cover the dish with foil or a lid, and bake in the preheated oven for 45 to 50 minutes, until the casserole is bubbling, the vegetables are tender, and the chicken is no longer pink inside. Uncover the dish and bake for another 10 to 15 minutes, until the vegetables are starting to brown. Let cool 5 minutes before serving.