Heat the oil in a skillet over medium-high heat. Rinse the chicken and pat dry with paper towels. Season the chicken with 1/8 teaspoon of the pepper and cook until golden brown on each side, 2 to 3 minutes. Transfer to a plate.
Add the onion and squash to skillet and cook, stirring occasionally, for 2 minutes. Add the broth, sage, and the remaining pepper and bring to a boil. Return the chicken to skillet. Transfer to oven and cook until the squash is tender and the chicken is cooked through, about 15 minutes.
Meanwhile, pour about 1/2 inch of water into a small saucepan and bring to a simmer over medium-high heat. Fit a steamer basket in pan. Place the broccoli in steamer, cover, and steam until just tender, about 3 minutes.
Transfer the chicken and vegetables to a plate and serve with the broccoli.