Mix butternut squash, brown sugar, and cinnamon together in a bowl.
Cook and stir andouille sausage and onion together in a stockpot until sausage is cooked through and onion is tender, 5 to 10 minutes. Add squash mixture, chicken stock, water, quinoa, cumin, and chicken bouillon to sausage mixture and bring to a boil.
Reduce heat to medium-low and simmer until quinoa is mostly tender, about 10 minutes. Add half-and-half and simmer until heated through, about 10 minutes.