Preheat oven to 425 degrees F. Lightly butter a casserole dish.
Put the squash on a rimmed baking sheet, drizzle with EVOO and season with salt, pepper and nutmeg. Toss to combine. Roast until the edges are browned, 17 to 20 minutes. Reduce the oven temperature to 400 degrees F.
Cook the pasta in boiling salted water to just shy of al dente, 7 minutes. Drain.
Meanwhile, melt the butter in a saucepan over medium heat. Add the chili powder, oregano, coriander, cumin, cinnamon, garlic and chile pepper and cook, stirring, for 2 minutes. Add the flour and cook, stirring, then whisk in the milk. Bring to a simmer and cook until the sauce is thick enough to coat the back of the spoon. Add the Cheddar and Parmigiano-Reggiano and stir until melted.
Combine the squash, chili-cheese sauce and beans, then transfer to the prepared casserole dish. Top with the pepper jack cheese. Cover and bake for 30 minutes. Uncover and bake until the top is browned, 15 to 20 minutes more.