You believe ?, well in this context, "believe" = don't know aka can't prove, i.e., your belief is speculation, possibly based on a basis of accurate fact.
His daughter said June 1849, that sounds pretty specific, and she would have been in a position to know the true facts, BUT, who came up with the name of the ship Ashland ?, her or you. ?
The following information is from a collection of passenger manifests for passenger arrivals over several years in the 1840's.
It was created by an independent academic research organisation and is available in an online searchable database provided by NARA the U.S. National Archives and Records Administration.
IF he landed in New York in 1849, then there are listings for 3 John Atkins, all Irish born, BUT only one of them corresponds to an arrival date of June 1849.
Passenger manifest details were pretty sparse back then, so ultimately, you might not be able to prove 100% that this is him, but at least it does match his daughter's facts.
John Atkins aged 30 born Ireland, arrived New York, from Liverpool, June 19th 1849, ship Josephine.
http://aad.archives.gov/aad/display-partial-records.jsp?dt=1...http://aad.archives.gov/aad/record-detail.jsp?dt=180&mtc...Start here in order to search by ship or by passenger name.
http://aad.archives.gov/aad/series-description.jsp?s=639&...