Homecoming Enhancement Research and Oversight Act - Directs the Secretary of Defense to enter into an agreement with the National Academy of Sciences for a study of the physical and mental health and other readjustment needs of members and former members of the Armed Forces who deployed in Operations Iraqi Freedom or Enduring Freedom, and their families. Requires: (1) reports, from the Academy to the Secretaries of Defense and Veterans Affairs and from the Secretary to Congress, on such study; (2) the public availability of the reports; (3) such Secretaries to develop a joint plan to address report findings; and (4) the public availability of the Secretaries' response.