Heat the olive oil in a medium stockpot set over medium heat and add the onion and salt. Cook until softened and slightly caramelized, about 3 minutes.
Add the butternut squash and cook for about 1 minute, then throw in the uncooked quinoa (make sure you rinsed it well first), chicken broth and water; bring to boil then simmer uncovered until the quinoa is completely cooked, about 15 minutes.
Kill the heat, add the cooked chicken and arugula. Give the soup a quick stir, cover and let it sit for a few minutes until the arugula is completely wilted and the chicken is heated through.