I am also looking for the Henry family from County Tyrone, Ireland. Hugh Henry (about 1837-1903) and his brother William Henry (about 1831-1914) emigrated to Australia in 1856.
According to their marriage and death certificates their father was John Henry and mother Margaret Warnock.
Hugh Henry born Newmills, Co Tyrone and William Henry born Malines, Co Tyrone.
Both records on the brothers show that John Henry was a farmer then publican.
Unfortunately I can not find any Irish records.