Cook the rice according to package directions. Drain off any excess cooking liquid and set aside.
Preheat your oven to 350 degrees F. Lightly coat a 9x13-inch baking dish with cooking spray and set aside.
In a large, deep skillet, heat 2 tablespoons olive oil over medium heat. Add the diced chicken and sauté until the chicken is cooked through and no longer pink on the inside, about 6 minutes. Remove to a paper towel-lined plate and set aside.
With a paper towel, carefully wipe the skillet clean. Heat the remaining 2 tablespoons olive oil over medium. Add the diced squash, onions, salt, and pepper. Sauté until the onion begins to soften, about 2 minutes. Cover and cook, stirring occasionally, until the squash is tender but still retains some chew, about 6-8 minutes. Stir in the thyme, cranberries, reserved chicken, rice, and 1/4 cup Parmesan.
Transfer the mixture to the prepared baking dish. Bake in the oven for 20 minutes, then remove from the oven and sprinkle with the remaining 1/4 cup Parmesan cheese. Bake 5 additional minutes, until the cheese melts. Sprinkle with additional chopped fresh thyme and serve warm.
To freeze: Bake the casserole completely, then let cool to room temperature. Cover it tightly, then freeze for up to 3 months. Let thaw overnight in the refrigerator, then reheat, covered, in a 350 degree oven until warmed through.
You could also try freezing the casserole unbaked. Again, let thaw overnight in the refrigerator, then bake as directed.