Can an illegal alien couple divorce in the United States?

No, persons who are unlawfully present in the U.S. do not qualify for traditional legal status, such as marriage and divorce. If said persons were married within the U.S. the marriage would not be considered valid in the legal sense and therefore a petition for the dissolution of the union would not be necessary. This does not apply when an undocumented foreign national marries a U.S. citizen.