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. Stir pasta, chicken, green bell pepper, eggs, Parmesan cheese, and red onion together in a bowl. Pour salad dressing over the top and toss to coat. Cover the bowl with plastic wrap and refrigerate until chilled and flavors blend, at least 2 hours.