In large pot, combine chicken broth, bouillon cube, rice, carrots, celery and onion. Bring to boil. Reduce heat, cover and simmer for 20 minutes. Stir in chicken.
In small pan, melt butter and stir in flour. Cook for 1 minute, stirring constantly. Gradually add in broth mixture and cook until thickened, stirring constantly.
In small bowl, beat eggs until frothy. Beat in lemon juice and 2 cups of thickened broth mixture. Slowly add egg mixture to broth mixture in large pot, stirring constantly. Heat gently until soup thickens enough to coat spoon. DO NOT BOIL.