Preheat the oven to 350°F. Set the squash quarters, skin-side down, in a 9" x 13" baking pan. Pour 1/2" of water into the dish. Cover the pan with foil and bake squash until tender, about 30 to 35 minutes.
Meanwhile, place chicken and cornstarch in a bowl and toss to coat. Place oil in a large skillet set over medium-high heat. When hot, cook the chicken, in batches, until nicely seared, about 2 minutes per side. Remove the cooked pieces of chicken from the pan as you go along and set on a plate.
When the chicken is cooked, add the onion, celery and ginger to the skillet and cook 2 minutes. Mix in the curry powder and cook 1 minute. Add the coconut milk, stock, lime juice and sugar. Return chicken to the skillet, bring to a slow simmer, adjusting the heat downward to maintain that simmer. Simmer 5 minutes. Add the apples and simmer 10 minutes more. Mix in the mint, season with salt, and then serve the curry with small bowls of the curry toppings.