Observations:
Insofar as I'm aware, no such object is shown in any book on civil war Military insignia or buttons or etc. If I'm incorrect about that, please let me know.
The one in the finders photo shows a Roman letter R on a rayed (starburst) background. No Confederate (nor yankee) Rifleman's emblem shows the letter R on a rayed background.
For those reasons, in combination, I am reasonably certain that the object is not a Military-issue object -- but instead, was manufactured strictly for Civilian usage.
Yes, it's not impossible that a soldier employed it on his uniform -- just as we know that they did so with Civilian flower-buttons and flatbuttons as a temporary replacement for a lost Military button.
At this point, I must shift (temporarily) into Creskol mode and say that it drives me up the wall to see so many flower-buttons and flatbuttons beng advertized on Ebay (and other websites) as being "Confederate buttons." And that's just mentioning only two such misrepresentations/"misdentifications." If it looks like it was made before 1920, it's a civil war relic. If it's old and round-ish and iron, by golly it must be a civil war Confederate cannonball. World War One crossed-sabers insignia and definitely-postwar Civilian horsegear - hey, I think they are Confederate Cavalry relics. Trust me that over the past several decades, Creskol and I have seen thousands of such claims. That number is not an exaggeration. So, I'm going to ask you to cut Creskol a lot of slack about his reaction to the suggestion that Relic Dude's "R" emblem is a civil war relic (which automatically means military in most people's mental interpretation). Trust me also that the longer y'all stay in this hobby, continuing to learn more and more about correctly identifying Historical artifacts, the more like Creskol (and me) you will become.
On that note... let me confess that one week ago I very nearly erupted like Creskol upon seeing a statement that an ironbacked civilian Fashion button in the What-Is-It forum is a Confederate Engineers button. Rigid self-control was required for me to reply with just: "if it really is a Confederate Engineers button (or any other version of Confederate or US military buttons), then please explain to us why it doesn't appear in ANY button-book which shows Confederate and/or US military buttons."
I guess it comes down to how many unproven suggestions you've seen in the past several days that a relic is a civil war one. I suspect that my fellow longtime civil war relic collector, digger, and ID-helper Creskol might have seen one too many such suggestions this week, and it was the proverbial straw that broke the camel's back. So again I'm asking y'all to cut him a goodly amount of slack. His point was valid, after all, although not gently spoken.
Now, back to the "R" object. Precise size-measurements by RelicDude would be best. But until he provides them, here is my analysis of its size from comparison with the penny in one of the photos. A penny is .75-inch in diameter. The rayed-background "R" disc seems to be about .7" in diameter, the smaller disc on its back is about 3/8-inch in diameter, and the distance between them is about 1/4-inch. A cufflink typically has about twice as much space (a half-inch) between the discs, so I'm going to discount RelicDigger's object from being a cufflink. I also note that the disc on its back isn't flat like a typical rivet's back. Instead, it is gently "domed" (to prevent it from chafing whatever it rubs against) -- so for the time being, I'm going to go with it being a collar-stud... one with a Family-name Initial on it.
Breezie
with you! If you read my first post I stated "It attached more of less too like a cuff-link, knapsack,leather strap or something in that nature (Military maybe?)! It appears very old maybe mid 19th-late 19th century? Can't say for sure if this is representing Riflemans or not but it sure looks like it could be." 
