It's a Henry cartridge, and yes they were used during the CW, mainly by the Northern Army, the Northern Army bought 1731 Henry rifles for use in the CW, but most of the Henry rifles used in the CW were not issued by the government, most of the Henry rifles that were used were usually bought by the soldier using them, the Southern Army did use the Henry if they capture one or a soldier bought one but it was hard for them to obtain ammo for the rifle.
One thing of note is that there were some pistols that also used the same ammo, but I don't think any of them had the double firing pin setup, maybe someone else could verify if that is the case.
.44 Henry - Wikipedia, the free encyclopedia
Henry rifle - Wikipedia, the free encyclopedia
Here's a link for some of the ammo that was used in Henry rifles, not all of the ammo had the H head stamp, the Winchester ammo and New Haven Arms Company was stamp with the H, other stamps were Remington stamped with a U, Dominion Cartridge Company stamped with a D, and United Sates Cartridge Company stamped with US, there were other companies that also product ammo for the Henry, some had no head stamp at all, early New Haven ammo didn't have a head stamp.
44Henry
Also if yo could determine if the cartridge is made of copper or brass, it would help date it, the first one made were made of copper and later brass.