I am unable to point the path yet to Co Down, so far trying to find the connection I rechecked for irish born oO'Neill's in Cumberland and in
www.familysearch.com found two entries of interest
John O NEILL Birth Year <1862> Birthplace Ireland Age 19 Occupation Soldier <Unmarried> Head of Household John POTTS
Source Information: Institution "2nd Brigade Dep The Castle Barracks" Carlisle .Census Place Rickergate, Cumberland, England .Family History Library Film 1342245 . Household: full of soldiers John O'NEILL Private 19 Ireland Soldier Source Information:Institution "2nd Brigade Dep The Castle Barracks" Carlisle Census Place Rickergate, Cumberland, England Family History Library Film 1342245
Also under same sourceI wonder if your family is the one with an entry in Family search
Head of Household Jane O'NEILL Source Information: Dwelling No 6 Aldbie St
Census Place Cleator, Cumberland, England .Family History Library Film 1342252 .Household Jane O'NEILL Head W Female 52 Scotland Thomas O'NEILL Son U Male 32 Scotland Iron Ore Miner. John O'NEILL Son U Male 17 Ireland Lab Iron Ore Pit Source Information:Dwelling No 6 Aldbie St Census Place Cleator, Cumberland, England Family History Library Film 1342252 etc etc you need to look up the original entry ec.
So after a small hunt in Co Down for Hugh Henry O’Neill and John O’Neill, I realised the older Brother Thomas was born in Scotland, could Hugh Henry O'Neill have also been born in Scotland and the family went to Ireland during the 1860s. Do you know anything more of a lasting Irish ancestry in Co Down connection or Scotland? Keep the hunt going Jim