Preheat oven to 425 degrees F (220 degrees C). Place the squash halves, cut-side up into a small baking dish, and bake until tender, about 45 minutes.
Bring the quinoa and water to a boil in a saucepan over high heat. Reduce heat to medium-low, cover, and simmer until the quinoa is tender, 20 to 25 minutes. When done, stir in the butter until melted, then season to taste with salt and pepper.
Meanwhile, heat the olive oil in a skillet over medium heat. Stir in the onion, green pepper, and celery; cook and stir until the onion has softened and turned translucent, about 10 minutes. Stir in the apple, and continue cooking until the vegetables are tender, about 10 minutes more. Add the garlic and ginger, cook 2 minutes more, then stir in the vinegar, sugar, cinnamon, nutmeg, and red pepper flakes; season to taste with salt and pepper.
When the squash is tender, scoop out some of the flesh, leaving the halves 1/2 inch thick. Roughly chop the acorn squash, and combine with the quinoa and apple mixture. Spoon the mixture back into the squash shells, and sprinkle with the mozzarella cheese. Return to the oven, and bake until the cheese has melted and is bubbly, about 5 minutes.