McCaffery

McCaffrey (and variant spellings) is a fairly common name in Ireland. In the 1901 census there were 1692, of which 138 were named John and 118 James. The names would have been equally common in the first half of the 1800s you’re your ancestors were born. Without parents names or some other way of narrowing the search I’d say you are looking for a needle in a haystack.

Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.

The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price.  http://www.nifhs.org (Go to DNA project on the website).

Enclosed is John McCaffery's Death Certificate naming his mother as Margaret..Sorry but that seems to be the limit of my Ireland/ McCaffery knowledge

(Maybe a Andrew McCaffery who  sponsored their migration to Australia ? )