Preheat oven to 350°F. Line baking sheet with foil and coat with cooking spray. Place squash cut-side down on prepared baking sheet, and tuck garlic cloves in hollow under squash. Bake 50 minutes, or until tender. Cool 10 minutes. Scoop flesh from shell, and place in large bowl. Peel garlic cloves.
Heat large pot over medium heat, and coat with cooking spray. Add onion, and cook 7 minutes, or until translucent. Add ginger, allspice and cinnamon, and cook 1 minute more. Stir in squash, garlic, broth and 3 & 1/2 cups water. Bring to a simmer, and reduce heat to medium low. Partially cover pot, and simmer 25 minutes. Purée in blender or food processor until smooth. Serve hot or cold, and stir in chives and vinegar just before serving.