Preheat oven to 350 degrees F (175 degrees C). Grease a 1-quart baking dish.
Melt 1/4 cup butter in a saucepan over medium heat. Cook and stir squash and onion in hot butter until tender, 10 to 15 minutes. Drain and cool vegetables.
Whisk sour cream and egg together in a large bowl. Add squash mixture, Cheddar cheese, and 1/2 of the crushed buttery round crackers; stir to combine and pour into prepared baking dish.
Stir remaining buttery round crackers and melted butter together in a bowl; spread over the top of squash mixture and gently pat down.
Bake in the preheated oven until crackers are lightly browned, about 20 minutes.