Preheat oven to 350. butter an ovenproof casserole dish.
Bring pot of salted water to a boil. Place yellow squash and onion in steamer basket and set over boiling water. Steam until squash is jsut tender. Repeat with zucchini, steaming until just tender. Transfer squash and onion to bowl. Add brown sugar, eggs , butter, half and half and cheese. Season with salt and pepper and stir to combine. Transfer to prepared casserole and bake until firm and brown on top, 30 to 45 minutes. Cool slightly before serving.