Heat a large stainless steel skillet over high heat. Sprinkle chicken evenly with 1/4 teaspoon salt and pepper. Add oil to pan; swirl to coat. Add chicken, skin side down; cook 3 minutes or until skin begins to brown. Remove chicken from pan. Pour oil into a small bowl; reserve oil. Return chicken to pan, skin side down. Place pan in oven; bake chicken at 425° for 4 minutes. Turn chicken over, skin side up, and bake an additional 2 minutes or until done. Remove chicken from pan; place on a cutting board. Let chicken stand, skin side up, 10 minutes. Cut chicken into slices; keep warm.
Cook the noodles according to package directions, and drain. Sprinkle noodles with 1/4 teaspoon salt; toss well to combine.
Combine 1/4 teaspoon salt, cilantro, and next 5 ingredients (through chiles) in a food processor, and process until well blended.
Return the skillet to medium-high heat. Add reserved oil to pan; swirl to coat. Add squash, bok choy, and remaining 1/4 teaspoon salt; sauté for 3 minutes or until vegetables are crisp-tender, stirring frequently. Stir in soy sauce. Divide noodle mixture evenly among 4 bowls. Top each serving with 1/2 cup squash mixture, 1 chicken thigh, and 2 lime wedges. Drizzle the sauce evenly over servings.