Preheat oven to 350 degrees F (175 degrees C). Grease a 15x17-inch baking dish.
Place the squash into a saucepan with about 1 cup of water, cover, and cook over medium-low heat until soft, about 15 minutes; drain, saving about 3/4 cup of the squash cooking liquid, and set aside. Break the cornbread apart into chunks, place into a large bowl, and chop up with a dinner knife until crumbled.
Melt the butter in a skillet over medium heat, and cook the onion, green bell pepper, and celery until soft, about 10 minutes, stirring often. Mix the cream of chicken soup and eggs together in a bowl; mix in the pan-fried vegetables, and season to taste with salt and black pepper. Stir the soup mixture into the cornbread. Gently stir in the cooked squash, and lightly mix in enough of the reserved squash cooking liquid to give the dressing the consistency of cooked oatmeal (slightly soupy). Stir in the cubed cheese food. Turn the dressing into the prepared baking dish.
Bake in the preheated oven until set and lightly browned, about 1 hour; let stand 10 to 15 minutes before serving.