Any chance this family is related to a John Ahern in Troy, New York? The following ad appeared in The Pilot, 26 March 1864.
"Of MATTHEW AHARN, of the city of Limerick; when last heard from he was in Oswego, NY in the year 1862. Any information of him will be thankfully received by his brother, John Aharn, 194 Fourth street, Troy, NY."
Another ad appeared 30 July 1864:
"Of MATTHEW AHEARN, of Rathmore, between Bruff and Fredmore, county Limerick. He left home when quite young, and was last seen at Troy, N Y seven years ago. An anxious widowed sister would be thankful for any information concerning him, and any one furnishing such information will confer an act of charity which will be deeply appreciated. Address Margaretta Ahearn, daughter of Mary and Timothy Ahearn, Washington, D C."
If you enter Troy in the search engine of the Ahern website you will get hits on 28 pages.