Using a large, sharp knife, pierce the spaghetti squash in several places.
Place the spaghetti squash in a glass baking dish and cook in the microwave on high for about 15 minutes, turning the squash halfway through cooking.
Before handling, let the squash stand for 10 minutes. Cut in half lengthwise and scrape out the seeds and fibers. Using a fork, twist out strands of the spaghetti squash flesh and place in a large bowl. Keep warm until ready to serve.
Lightly coat a large nonstick skillet with cooking spray and set over medium-high heat. Add the chicken, season with salt and pepper, and cook until just cooked through. Remove from the pan and set aside.
Heat the olive oil in the skillet set and turn the heat to medium. Add the garlic, anchovies, oregano and red pepper flakes, and cook for 30 seconds to 1 minute, breaking the anchovies down with the back of a wooden spoon. Avoid burning the garlic.
Stir in the crushed tomatoes, kalamata olives and capers. Bring to a boil, then simmer for about 5 minutes.
Stir in the cooked chicken and parsley, season with salt and pepper if necessary and serve.
Divide the spaghetti squash between 4 plates and top with the sauce. Serve.