Combine turmeric, 1/2 tsp. salt, 1 tsp. allspice, and 1/8 tsp. cayenne in a large bowl. Add chicken and toss to coat. Arrange chicken skin side up alongside squash and onions on baking sheet. Brush tops of chicken with 1 Tbsp. olive oil. Bake until interior of squash and onions are fork-tender, chicken skin is crispy, and an instant read thermometer inserted into the thickest part of the chicken registers 165°F, 35–45 minutes; if chicken is done before squash, remove from oven and let rest until squash is finished cooking.