Cook pasta in a large saucepan as directed on package.
Meanwhile, sprinkle chicken with salt and pepper. Heat 1 TBSP oil in a large nonstick skillet on medium heat. Add chicken; cook 5 to 7 minutes on each side or until done. Remove to plate and cover to keep warm. Add remaining oil in the same skillet, add squash and zucchini, sautee for 5 minutes. Add garlic and onion, fry down for 3 more minutes or until squash is crisp-tender.
Drain pasta, reserving 1/4 cup of pasta water. Toss pasta in same cooking pot with the reserved water, place pasta on a large serving platter, top with chicken mixture, cheeses and basil.