Trim the stem and root end off the squash and slice it in half lengthwise. Slice each half in half again. Scrape a spoon along down the length of the pieces to remove the seeds and cut into ½-inch wide chunks. Put the squash in a large mixing bowl.
Add olive oil, salt, shallots, garlic and chili to the squash and toss together. Transfer to a rimmed baking sheet in a single layer and roast 20-25 minutes. The squash should be tender and nicely brown. Drizzle the balsamic over the squash as soon as it comes out of the oven and stir around gently with a spatula.
While the squash is roasting, bring a large pot of salted water to a boil. Cook the rigatoni until al dente. Before draining the pasta, add the greens to the colander. The brief contact with the boiling pasta water will wilt the greens. If you're using a tender green like arugula, just toss it in at the end.
Transfer the pasta and greens back to the cooking pot and stir in the cream and Parmesan.
Add the squash to the pasta and toss together. Serve hot with additional cheese and chili to your taste.