Preheat the oven to 350 degrees F.
In a heated skillet, cook the onions, adding a little water to prevent them from burning. Cook until lightly brown, adding more water, if needed. Add the squash, season with salt and pepper, to taste, and cook until the squash is tender, about 8 to 10 minutes. Remove the skillet from the heat and stir in the mayonnaise, 1 cup of the cheese and 1 cup of the crumbled crackers. Pour the mixture into a 1 1/2-quart baking dish. Sprinkle with remaining cheese and top with remaining 1/4 cup of butter crackers. Bake until the casserole is bubbly around edges, about 25 minutes. Remove the casserole from the oven and serve hot.