Preheat the oven to 450 degrees F. Toss the squash, sliced lemon, 2 tablespoons olive oil, 1 tablespoon oregano, 1/2 teaspoon salt and a few grinds of pepper in a large bowl. Spread on a rimmed baking sheet. Roast, undisturbed, until the squash is tender and lightly browned around the edges, about 30 minutes.
Meanwhile, season the chicken with salt and pepper. Heat a large ovenproof skillet over medium-high heat. Add the remaining 2 tablespoons olive oil. Add the chicken; cook until well browned on the bottom, 3 to 5 minutes. Flip the chicken and transfer the skillet to the oven. Roast until the chicken is just cooked through, 5 to 8 minutes.
Return the skillet to the stovetop over medium heat. Add the garlic and cook until softened, about 1 minute. Add the lemon juice and the remaining 1 tablespoon oregano. Cook, spooning the pan juices over the chicken, 1 minute. Serve the chicken with the roasted squash and lemon; drizzle with the remaining pan juices.