The subtlety of leeks and shallots complements the bold flavor of cremini mushrooms in this rich stuffing.
Arrange bread in a single layer on a baking sheet, and let stand, uncovered, at room temperature overnight. (Alternatively, bake bread in a 250-degree oven until dry but not browned, 25 to 30 minutes.)
Transfer bread to a large bowl. Preheat oven to 400 degrees.
In a medium bowl, toss squash with 2 tablespoons olive oil until coated; season with salt and pepper. Arrange squash in an even layer on a baking sheet; transfer to oven and roast until squash is tender, about 15 minutes. Transfer squash to bowl with bread cubes.
Heat 2 tablespoons oil in a large skillet over medium-low heat. Add shallots, celery, and leek. Cook, stirring occasionally, until vegetables are soft, about 5 minutes. Fold in mustard greens and cook until just wilted, about 1 minute. Transfer to bowl with bread and squash.
Heat remaining 2 tablespoons oil in skillet over medium heat. Add mushrooms and cook, stirring occasionally, until softened, 2 to 3 minutes. Transfer to bowl with bread and vegetables.
Heat chicken broth in a small saucepan with butter and over medium-low heat until butter is melted; pour over bread mixture. Add sage, parsley, and thyme to bread mixture; season with salt and pepper; mix until well combined. Let stuffing mixture cool completely if stuffing in turkey.
To cook stuffing in a turkey: Loosely stuff into a turkey, and roast until center of stuffing registers 165 degrees on an instant-read thermometer. To bake all of the stuffing outside a turkey: Spoon into a buttered 2-quart baking dish. Cover dish with parchment paper-lined aluminum foil. Bake at 400 degrees for 20 minutes. Uncover and bake until golden brown, about 25 minutes more.