Thank you very much for your query to Ireland Reaching Out. You should post your message on the Ballina message board. Do you know if this was Ballina, Co. Mayo or Ballina, Co. Clare? Please post it on the appropriate county, as there may be someone in the community there, who might know more information relating to Mathias Breen. Also post it on the Kilfintinin parish page: http://www.irelandxo.com/group/kilfintinan-clare
Clare County Library have an excellent resource list for County Clare and you might be able to find the grave inscriptions you mentioned in Killofin there.
Do you know the name of the ship Mathias emigrated on?If you do, there might be details available in the US. Usually this information was detailed in the port of arrival rather than the port of departure.
You should try getting in touch with the parish priest in the area. They might have more details from church registers. Or the General Register Office (GRO) for records of births etc.
You have done very well in your search so far. Best of luck!