Place the butternut squash and a splash of water in a large bowl with a lid in the microwave, and cook until barely tender, about 6-7 minutes. Remove half of the squash and set it aside. Cook the remaining squash an additional 3-4 minutes or until it is very tender. Drain, then mash this squash with the back of a fork and also set aside.
Add the chicken broth and chicken breasts to a large soup pot on the stove, and bring to a boil. Place a lid on top, lower the heat to medium-low, and simmer for 10 minutes or until chicken is cooked all the way through. Remove chicken to a plate to cool, and pour chicken broth into a separate bowl.
Return the pot to the stove over medium-low heat, and add in the olive oil. Add onion, salt, and pepper and saute until the onion is soft and brown, about 10 minutes. Add garlic and oregano. Cook, stirring constantly, for another minute.
Add tomatoes and all the butternut squash to the pot. Stir to combine. Pour in reserved chicken broth and quinoa, and stir to combine. Bring to a boil then place a lid on top, lower the heat to medium-low, and simmer for 15 minutes or until quinoa is tender and cooked.
Meanwhile, shred the chicken. When quinoa is cooked, stir in chicken, olives and parsley. Cook until chicken is hot, season with additional salt & pepper if necessary, then ladle into bowls.
Notes
Keep extra chicken broth on hand for reheating as the quinoa will soak up some of the broth.