Heat oven to 450° F with the racks in the upper and lower positions.
Toss the squash, 2 tablespoons of the oil, and ¼ teaspoon each salt and pepper on a rimmed baking sheet.
Place the chicken on a separate rimmed baking sheet and season with ¼ teaspoon each salt and pepper.
Roast the chicken on the top rack until an instant-read thermometer inserted in the thickest thigh (avoiding the bone) registers 165° F, 20 to 25 minutes.
Roast the squash on the bottom rack (do not toss) until golden brown and tender, 8 to 10 minutes.
Toss the squash, radishes, scallion, lemon zest and juice, the remaining tablespoon of oil, and ¼ teaspoon each salt and pepper in a medium bowl.
Serve the chicken with the squash salad.