In medium bowl, mix cheese, chicken, 1/3 cup salsa and the squash purée.
Spray 1 side of 1 tortilla with cooking spray; place sprayed side down in 10-inch skillet. Spread with one-fourth of the chicken mixture. Top with another tortilla; spray top of tortilla with cooking spray.
Cook over medium heat about 2 minutes or until golden brown. Carefully turn quesadilla over; cook 1 to 2 minutes or until golden brown. Repeat with remaining tortillas and chicken mixture. To serve, cut quesadillas into wedges. Serve with sour cream and additional salsa.