Place chicken in a 6-qt. stockpot; add water to cover. Slowly bring to a boil. Reduce heat; simmer, covered, 2-3 hours. Meanwhile, in a large bowl, mix 1-1/2 teaspoons salt, oregano, basil and 1 teaspoon pepper. Add beef; mix lightly but thoroughly. Shape into 1/2-in. balls.
Remove carcass from stockpot; cool. Return broth to a simmer; add meatballs. Cook, uncovered, 8-10 minutes or until meatballs are cooked through.
Remove meat from carcass; shred meat with two forks and return to pot. Discard carcass and skin. Bring broth to a boil; stir in rice and spinach. Reduce heat; simmer, covered, 5 minutes. Drizzle beaten eggs into soup, stirring constantly. Stir in remaining salt and pepper.