Yogurt provides a tangy contrast to the sweetness of the apples and the cider; squash puree gives the soup a creamy texture, without the fat of heavy cream.
In a large pot (6 to 8 quarts), heat oil over medium heat. Add onion; cook, stirring frequently, until beginning to brown, 4 to 6 minutes. Stir in apples; cook, stirring occasionally, until softened, 3 to 5 minutes. Remove 1/2 cup apple mixture, and set aside for garnish.
Stir cider, ginger, and 2 1/2 cups water into apple mixture in pot; bring to a boil. Stir in squash puree; cook until thawed, 8 to 10 minutes. Reduce heat; simmer until mixture has thickened, 15 to 20 minutes. Season with salt and pepper.
Ladle soup into bowls. Garnish each serving with 1 tablespoon yogurt and 2 tablespoons of reserved apple mixture.