To make it...
Heat oven to 325ºF. Brush each squash wedge with about 1 teaspoon olive oil, then sprinkle with about 1/4 teaspoon salt and a pinch of cayenne if using. Place the squash quarters on a baking sheet skin-side down. (Alternatively, arrange on the rack with the turkey.) Roast during the turkey's last 45 minutes in the oven. The squash is done when it is fork-tender. Meanwhile, grate the zest (the orange skin, not the white pith) and squeeze the juice from the orange. Remove the squash from the oven and immediately sprinkle with the mint, orange juice, and zest.
To fake it...and save 35 minutes
Prepare three 12-ounce packages frozen winter squash according to the label directions. With a wooden spoon, beat in the grated zest of 1 orange, 3 tablespoons butter, 2 tablespoons chopped fresh mint leaves, 1 tablespoon sugar, 1 teaspoon salt, and 1/4 teaspoon cayenne (optional).