Preheat the oven to 375 degrees F and butter a 3-quart baking dish. Melt 2 tablespoons butter in a large heavy-bottomed pot over medium heat. Add the mushrooms and cook, stirring occasionally, until all the moisture evaporates and the mushrooms start browning, about 7 minutes. Season with salt and pepper. Remove the mushrooms to a plate.
Melt the remaining 6 tablespoons butter in the pot, then add the squash, collards, onion, celery, sage, thyme, 1 1/4 teaspoons salt and a few grinds of pepper. Cook, stirring, until the vegetables are tender, about 10 minutes. Add the broth and bring to a boil, then remove from the heat.
Whisk the eggs and parsley in a large bowl. Add the bread, mushrooms and vegetable-broth mixture and stir until combined. Transfer to the prepared baking dish. Drizzle with the turkey drippings (or dot with butter).
Cover the dish with foil and bake 30 minutes, then uncover and bake until golden, about 30 more minutes.
You can assemble this stuffing the day before - keep in the fridge until baking.