Combine the squash, kale, 1 tablespoon butter, 1/2 teaspoon salt, 1/4 teaspoon pepper and 1/2 cup water in a large nonstick skillet; bring to a boil over medium-high heat. Cover and simmer until the squash is tender, 5 to 8 minutes. Uncover and cook until the water is evaporated and the squash is beginning to brown, about 5 minutes. Transfer to a bowl.
Heat 3 tablespoons butter in the same skillet over high heat. Add the mushrooms and hominy and cook, stirring once, until browned, 4 to 6 minutes. Reduce the heat to medium high, season with salt and pepper and cook, tossing frequently, until the mushrooms are tender, about 2 minutes. Transfer to the bowl with the squash mixture.
Heat the remaining 3 tablespoons butter in the skillet over medium-high heat. Add the red onion, bell pepper, chile pepper, garlic, thyme and a pinch each of salt and pepper. Cook, tossing, until the vegetables are tender and beginning to brown, 5 to 8 minutes. Add the corn and season with salt and pepper; cook, tossing, until the corn is tender, 6 to 8 minutes. Add the squash mixture to the skillet and toss. Transfer to a serving dish.
Make the succotash up to 4 hours ahead. Transfer to a shallow baking dish and gently reheat in a 300 degrees F oven.