Adjust oven rack to upper-middle position and heat oven to 475 degrees.
Combine oil, sage, salt, and pepper in bowl.
Combine 1 tablespoon of the sage oil with yogurt, water, and orange zest in separate bowl. Season with salt and pepper to taste and refrigerate yogurt sauce until ready to serve.
Cut each squash half into 2½- to 3-inch pieces. Combine squash and 3 tablespoons sage oil in large bowl; set aside.
Stir paprika into remaining sage oil. Pat chicken dry with paper towels. Using your fingers, gently loosen skin covering each breast piece, then rub paprika-sage oil evenly under skin.
Spread squash in single layer over three-quarters of rimmed baking sheet. Place chicken pieces skin side up on empty portion of sheet and roast for 15 minutes.
Meanwhile, vigorously squeeze and massage kale with hands in now-empty bowl until leaves are uniformly darkened and slightly wilted, about 1 minute.
Rotate sheet, stir kale into squash, and roast until vegetables are browned and tender and chicken registers 160 degrees, 10 to 20 minutes, stirring vegetables halfway through roasting.
Toss vegetables with any accumulated chicken juices. Serve, passing yogurt sauce separately.
