Melt butter in medium saucepan over medium heat. Add green onions; cook and stir about 3 minutes or until tender.
Stir in broth and rice; bring to a boil over medium high heat. Reduce heat to low; cover and simmer 20 to 25 minutes or until rice is tender.
Beat eggs in medium bowl. Stir in lemon juice and 1/2 cup broth mixture until blended. Gradually pour egg mixture into broth mixture in saucepan, stirring constantly. Cook and stir over low heat 2 to 3 minutes or until soup thickens enough to lightly coat spoon. Do not boil. Stir in pepper if desired.