Boil wild rice with its 3 cups water in saucepan. Cover; simmer on low 45 to 55 minutes until tender with some grains split. Drain excess.
Meanwhile, heat oven to 425°F. Toss squash with 2 teaspoons oil on rimmed sheet; roast 25 to 30 minutes, turning once, until tender with browned edges.
Heat remaining oil in 6-quart soup pot; cook onion 5 minutes. Add garlic and sage 30 seconds, then broth, salt, pepper, and squash. Simmer 5 minutes.
Immersion-blend smooth, blade submerged. Add cooked rice and turkey; gently simmer 5 to 8 minutes until turkey 165°F.
Add half-and-half; gently warm 2 minutes without hard boiling. Gradually add reserved broth if needed for pourable soup; serve.