In a 6-qt. stockpot, cook pasta according to package directions., In a large cast-iron or other heavy skillet, heat 1 tablespoon butter over medium heat. Add chicken; cook and stir until no longer pink, 7-9 minutes. Add 1/4 teaspoon each salt and pepper; remove from pan. In same pan, heat 1 tablespoon butter over medium heat. Add squash; cook and stir until tender, 3-5 minutes. Remove from heat., In a small saucepan, melt remaining 3 tablespoons butter over medium heat. Stir in flour and garlic until blended; gradually whisk in milk. Bring to a boil, stirring constantly until thickened, 1-2 minutes. Remove from heat; stir in cheese and the remaining salt and pepper., Drain pasta; return to pot. Add chicken, squash and sauce; heat through, stirring to combine.