Bring a large pot of salted water to a boil. Warm oil in a large nonstick skillet over medium-high heat. Add squash, and season with salt and pepper. Lower heat to medium and cook squash, stirring often, about 5 minutes. Add garlic and continue to cook until squash is tender, 5 to 7 minutes more.
Cook pasta according to package directions. Drain pasta, reserving 1 cup cooking liquid, and return pasta to cooking pot. Add chicken, spinach and half of reserved pasta water to skillet with squash. Cook, stirring occasionally, until heated through, about 2 minutes. Turn off heat and transfer squash mixture to pot with pasta. Add ricotta and Parmesan and toss. Pour in more reserved cooking liquid if mixture seems dry. Serve immediately.