Place squash and onion in a large skillet over medium heat. Pour in water. Cover, and cook until squash is tender, about 5 minutes. Drain well, and place in a large bowl.
Mix cracker crumbs and cheese together in a medium bowl. Stir half of the cracker mixture into the cooked squash and onions.
Mix eggs and milk together in a small bowl, then add to squash mixture.
Stir in 1/4 cup melted butter, and season with salt and pepper. Spread into the prepared baking dish. Sprinkle with remaining cracker mixture, and dot with 2 tablespoons butter.
Bake in the preheated oven until lightly browned, about 25 minutes.