Grease a 13-by-9-inch baking dish with unsalted butter. Set it aside.
Preheat the oven to 400°F. Add the cubed butternut squash to a baking sheet. Drizzle the butternut squash with olive oil, and roast the squash until browned and tender, about 30 minutes.
In a large nonstick pan set over medium heat, cook the sausage, using a spatula to break it apart into crumbles. Remove it to a paper towel-lined plate and drain any drippings from the pan.
To the same pan the sausage was cooked in, add the butter. Over medium heat, melt the butter then add the diced onions and cook, stirring occasionally, until they are softened, 3 to 5 minutes.
Transfer the cooked onions to a medium bowl then stir in the cooked sausage, roasted squash, fresh sage and cubed bread. Stir together the mixture then transfer it to the greased baking dish.
In a large bowl, whisk together the eggs, milk, salt and pepper.
Pour the egg mixture over the layers of bread and sausage. Cover the dish securely with plastic wrap and refrigerate it overnight, or for about 8 hours. (See Kelly's Note.)
Remove the strata from the fridge, discard the plastic wrap, and let it rest at room temperature for 30 minutes.
Preheat the oven to 350ºF. Place the strata on a baking sheet and bake it, uncovered, for 35 to 45 minutes, or until the eggs are fully cooked. Let the strata rest for 5 minutes before slicing and serving.