There is only one surviving green flag for an Irish CS unit and that is the 10th Tennessee Infantry. Please note, and this is a mistake that most folks make, it is not green on both sides - rather it is a 2 sided flag with the green on one side and a First National on the other. We have this flag in the Tennessee State Museum where I am doing the book on TN flags. Interestingly enough their first colonel, Adolphus Heiman, was a Prussian!
I also have a mention of another two sided Irish flag taken during the war. This was at the Battle of Belmont and it was taken from the pole over the CS camp on the west bank of the Mississippi River when Grant's forces charged into the camp after driving the Confederates from their line before it. This was the flag of Beltzhoover's LA Battery, an Irish unit from New Orleans, and it was taken by the Federals and has been lost ever since.
Another largely Irish unit was the 2nd Tennessee Infantry (Walker's) which came from Memphis. This is a different regiment than the 2nd Tennessee Infantry (Bate's) that came from Middle TN. The 2nd TN (Walker's) and the 21st TN were merged into the 5th Confederate infantry due to casualties after Shiloh and fought as that command for the rest of the war. At no time did they ever use any Irish flag that I have ever seen and I am not sure how many, if any, of the 21st TN was Irish. The 5th Confederate's Hardee pattern flag issued to them in 1864 rests in the TN State Museum as well. They fought for much of the war with Cleburne and would have carried versions of the Hardee flag accordingly.
The "Gods And Generals" movie tried to make the 24th Georgia into an all-Irish regiment showing their First National with the Harp of Erin on it. Only one company of this regiment was and that was the one rasied by its colonel initially. Their company flag served as the basis for the one in the film.