Cut squash lengthwise in half; discard seeds and stringy membrane. Cut each half lengthwise into quarters. Place squash and turnips on a baking sheet coated with cooking spray. Bake at 400° for 25 minutes; cool. Scoop out pulp from each squash piece; discard rind. Cut into 1-inch cubes.
Heat olive oil in a Dutch oven over medium heat. Add chopped onion, carrot, green onions, and garlic, and sauté 8 minutes. Add squash, turnips, chestnuts, broth, and tomatoes. Cover and simmer 25 minutes.
Combine cornstarch and water in a small bowl. Add cornstarch mixture, turkey, and remaining ingredients to pan; bring to a boil. Reduce heat; simmer 25 minutes.