Preheat oven to 450 degrees.
On a large baking sheet, toss together olive oil, onion, squash, garlic, salt and pepper; spread mixture evenly on baking sheet.
Bake for 20 minutes then add kale and basil; cook for 8 to 10 minutes or until squash is fork-tender.
Meanwhile, cook pasta according to package directions.
Transfer baked squash mixture to a large bowl; add hot pasta then toss with cheeses.