John O'leary. probably John A. O'leary

Based on my own searching experience in West Cork, and especially since you have a fairly narrow time frame to search, I suggest searching for baptisms in the relevant year(s) in the parish registers of West Cork, which are available online at https://registers.nli.ie , which has an interactive map to help you move from parish to parish.   The O'Leary surname originated in West Cork, so unfortunately there are many of them spread across the parishes there and on into Kerry (with some spillover also up into Limerick), but if you have to start somewhere, I suggest starting with the parish of Inchigeelagh/Iveleary (many parishes have more than one name), because the surname is said to have originated in or near that parish, and the name Iveleary is said to be based on the name Leary.  Then, try the parishes to the east and north of there next, because Leary's are very common there.

One technique I've learned is to find likely prospects, then check the registers further (although it can be quite time-consuming) to see whether they later married or had children in the parish, which would then rule them out for your family (unless they then emigrated with the family, of course, but in that case you'd usually know more about the family).  It's not an exact science, but you're lucky as to your time frame, because the birth would be post-famine.  Most of my Irish great-grandparents were born not long before the famine, so I could never be sure whether a child's absence from the register later on as an adult meant that they had emigrated or that they had died as a child (or the family moved elsewhere in Ireland during the famine/plague years).  What helped me was that I had the names of both parents in each case, and sometimes also the names of siblings, but I'm still looking for one great-grandparent.

I agree with Roger that, since the names you have are very common, DNA testng may be a good route, and might help to confirm that you've found the right records at some point.  I was lucky to locate the baptism record in Cork for my great-grandmother (a Cronin - another surname which origintaed in that area) in a parish a bit north of Iveleary, but I had the names of two of her siblings and both of her parents to help me.  I was then able to trace the family back into the late 1700's, but the interesting point is that I've since connected with several Cork relatives independent of that, through my DNA test results.   Some of them have Leary's in their family trees, by the way, and I probably also have some myself, though I haven't established any specific connections, apart from one Leary who was a sponsor at the baptism of one of my relatives.

If you do get DNA testing done, then (no matter which company you use) you shold also upload the results to the (free) GEDMATCH site:  https://www.gedmatch.com/login1.php where they can be compared to results from the other testing companies.  There are discussion forums at that site, and there's a partcular one for West Cork and Kerry which is very active.

Thanks for these replies. I will retry looking at the Irish genealogy and try some of the suggestions. I knew looking fo a John O'Leary in Cork would be an outside shot. Like I said, my grandmother would not talk about her family and her sister was equally traumatized. Even the Chicago records are very minimal and I will try writing the parish (es) that may have more. I will try the DNA route one of these days although my younger brother had one a year ago Christmas. Maybe I will see if he will upload the results also. Sorry for the delay. I mostly had, taught, college courses in history and poli. sci. on the intenet and I still had a little trouble navigating this site. The new reply boxes will help.