Preheat the oven to 350 degrees F (175 degrees C). Grease a 9x13-inch casserole dish. Combine zucchini and squash in a large pan, add water to cover, and boil for 5 minutes over medium-high heat. Drain and gently press zucchini and squash between paper towels to remove excess water. Meanwhile, melt 2 tablespoons butter in a skillet. Add onion and saute for 2 minutes. Add garlic and cook until onion has softened and garlic is fragrant, 2 more minutes. Mix in zucchini and squash. Add whole kernel corn, cream-style corn, Cheddar cheese, 1/2 cup Asiago cheese, 1/2 cup breadcrumbs, mayonnaise, eggs, salt, and pepper. Pour into the prepared casserole dish. Melt remaining 2 tablespoons butter in a skillet. Add remaining 1 cup breadcrumbs and remaining 1/2 cup Asiago cheese. Mix well and sprinkle on top of the casserole. Bake in the preheated oven until casserole is golden and set and cheese is melted, 45 to 50 minutes.