Preheat oven to 450°.
Toss the squash and onion in a large bowl with 2 tablespoons oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Spread in a single layer on a nonstick baking sheet, and bake 25 minutes, turning until tender and browned.
Cook the pasta according to package directions until al dente. Place chard in a large bowl. Drain the pasta in a sieve over the bowl of chard; let stand 1 minute. Drain chard well; squeeze out excess water. Toss with pasta in a large bowl along with roasted squash and onion, remaining 2 tablespoons oil, remaining salt and pepper, sage, and bacon. Top with cheese and serve.