If you each had a qualifying child AND you each supported your own households AND you did not live together at all the last 6 months of the year THEN you could each file as HOH ... the school is wrong.
BUT if you did live together for as little as ONE night the last 6 months of the year (hello honeymoon) then you must file either MFJ OR MFS so you need to amend 1 return and switch from HOH to MFJ OR you both need to amend from HOH to MFS.
