1. Preheat oven to 375 degrees Fahrenheit.
2. Place the acorn squash, sweet potatoes, apple, shallots, red onion, and garlic in a 9- x 13-inch baking pan.
3. Place the chicken quarters on top and season with garlic powder, onion powder, coriander, salt, cinnamon, and pepper. Drizzle the olive oil and vinegar on top and, using your hands, massage the spices into the chicken and vegetables.
4. Bake uncovered at 375 degrees Fahrenheit for 45 minutes. Reduce the heat to 350 degrees Fahrenheit, cover, and bake for another 75 to 80 minutes, or until the chicken is cooked through and the liquid bubbles. Serve hot.