Preheat oven to 475°.
Combine squash, vinegar, oil, and 1/4 teaspoon salt in a large bowl; toss well to coat. Arrange squash mixture in a single layer on a jelly-roll pan coated with cooking spray. Bake at 475° for 25 minutes or until tender and lightly browned, stirring occasionally.
While squash bakes, cook pasta according to the package directions, omitting salt and fat. Drain in a colander over a bowl, reserving 1 tablespoon cooking liquid.
Melt butter in a large nonstick skillet over medium heat. Add pine nuts, sage, and garlic; cook 3 minutes or just until the pine nuts begin to brown, stirring occasionally. Place pasta, reserved cooking liquid, pine nut mixture, and squash mixture in a large bowl; toss gently to combine. Add remaining 1/4 teaspoon of salt, arugula, cheese, and black pepper; toss gently to combine. Serve immediately.