In the 1860 census, there is a listing for John H. Berry, living 3rd Ward Jersey City, with probable spouse Eliza and daughter Christina. Possibly too old to be your John H, as he was born about 1797. Perhaps his father?
There is a John Barry, wife Mary, son Richard and daughter Hannah in the 1860, living Ward 2 Jersey City - born about 1835 in Ireland.
In the 1870, John M Beers, wife Mary, children Richard, Robert, Susan and William living Ward 2 Jersey City. Listed as a ship's carpenter.
Also in 1870, "John Been" in Ward 7 Jersey City, wife Marie, daughters Elizabeth and Kathy.