Preheat the oven to 350°F.

In a pot, boil the squash and onion in water to cover until just tender, about 12 minutes. Drain in a colander in the sink and press the vegetables to extract as much liquid as possible.

Melt 8 tablespoons (1 stick) of the butter in a small skillet. Add the flour and stir until smooth. Add the milk gradually, whisking constantly. Once the mixture gets hot, after about 5 minutes, it will thicken; it should be thick enough to coat the back of a spoon. Remove from the heat.

In a large bowl, combine the drained squash and onion, sauce, cheddar, parmesan, and salt and pepper to taste. Mix in the eggs. Pour into a 9 x 13-inch glass baking dish. Cut the remaining 4 tablespoons butter into six pieces. Top the casserole evenly with the cracker crumbs and dot with butter. Bake until the cracker crumbs are golden and the casserole is bubbling, 35 to 45 minutes.