Bring a large pot of lightly salted water to a boil; add penne and cook, stirring occasionally, until tender yet firm to the bite, about 7 minutes. Drain and rinse with cold water.

Transfer pasta to a large bowl. Stir in chicken, bell pepper, Parmesan, onion, and eggs. Pour in salad dressing and toss to coat. Cover with plastic wrap and refrigerate until chilled and flavors have blended, at least 2 hours.