Heat up the oil over medium high heat in a skillet. Add the honeynut cubes in a single layer and sear, stirring once or twice, for 5 minutes. Lower the heat to medium and cook until fork-tender, about 5-6 minutes, stirring as needed. Season with salt and pepper.
Meanwhile, cook the pasta 2 minutes shy of al dente in a large pot of salted water. Reserve 3/4 cups water and drain.
When the squash is done cooking, scoop the squash out of the pan and set aside. Add the cream and garlic to the same pan over medium heat and simmer until slightly reduced, 2-3 minutes.
Add the pasta and 1/2 cup pasta water to the sauce and bring to a boil, cooking, stirring occasionally, until the pasta is al dente and the sauce coats the pasta, about 2 minutes. Add more pasta water if the sauce starts to get too thick.
Stir in the greens until wilted, then turn the heat down and stir in the cheese until melted.
Add the honeynut squash, season with salt and pepper and enjoy immediately!