Cut the duck into 8 pieces: 2 leg/thigh portions, 2 wings, and a quartered breast with bone attached. In a large heavy pot or Dutch oven heat a 2-count of olive oil (about 2 tablespoons) over medium heat. Season duck pieces with salt and pepper. Carefully add them to the pan and cook for 8 to 10 minutes, until golden brown all over. Add coconut milk, star anise, cinnamon, and ginger to the pot; cover and simmer for 1 hour. Add the squash and stir. Cover and cook for 30 minutes more, until the duck is tender and the squash is cooked. Remove 1 cup of the braising liquid.

In a blender combine the 1 cup braising liquid, the green curry paste, cilantro, and lime juice. Puree until a bright green mixture forms. Fold paste back into the pot of curry and season with salt and pepper to taste. Serve with steamed rice, lime wedges, and fresh mint.