Bring Our Heroes Home Act This bill: (1) requires the National Archives and Records Administration (NARA) to establish the Missing Armed Forces Personnel Records Collection; and (2) establishes the Missing Armed Forces Personnel Records Review Board, which shall require all missing Armed Forces personnel records to be transmitted to NARA and disclosed to the public in the collection. Each government office shall: (1) identify, review, and organize each such record in its possession for transmission to NARA; and (2) identify and review for public disclosure each such record previously transferred to NARA that remains classified. Each presidential archival depository shall: (1) have as a priority the expedited review for public disclosure of such records in its possession, and (2) make such records available to the board. NARA shall: (1) make each such record identified as classified available for review by the originating body, and (2) prepare and make available to all government offices a standard form for collecting information relating to such records. All such records shall be publicly disclosed in full and made available in the collection within five years after enactment of this bill unless the President certifies that: (1) postponement is necessary because of an identifiable harm to the military defense, intelligence operations, or conduct of foreign relations; and (2) the harm is of such gravity that it outweighs the public interest in disclosure. The Board shall: (1) render a decision on a determination of a government office to postpone the disclosure of any such record; (2) determine, within two years after enactment of this bill, whether all government offices have complied with it; (3) terminate after four years; and (4) transfer all of its records to NARA for inclusion in the collection. The Board may request the Attorney General to petition any U.S. or foreign court to release information relevant to the loss, fate, or status of missing Armed Forces personnel that is held under seal of such court.