Preheat oven to 350°F. Grease an 11- x 7-inch (2 quart) baking dish with cooking spray.
Melt 3 tablespoons of the butter in a large skillet over medium-high. Add squash, onion, and 1 teaspoon of the salt; cook, stirring often, until center of squash is just tender and liquid has evaporated, about 10 minutes.
Transfer mixture to a colander set over a bowl. Drain 5 minutes; discard any liquid.
Stir together eggs, sour cream, Cheddar and Swiss cheeses, mayonnaise, thyme, pepper, and remaining 1 teaspoon salt in a large bowl. Gently fold in squash mixture.
Spoon into a lightly greased 11- x 7-inch (2-quart) baking dish.
Microwave remaining 3 tablespoons butter in a medium-size microwavable bowl on HIGH until melted, about 25 seconds. Toss together crackers, Parmesan cheese, and melted butter until combined.
Sprinkle topping over casserole.
Bake in preheated oven until golden brown, about 20 minutes.
