Whitney (Eli) made several military rifles, one of the main Civil War issues was the Model 1841 Mississippi rifle originally in 54 caliber, some were war-time converted to
standard 58 caliber and fitted with bayonet adapters. This was an amazingly accurate weapon. My guess is the Whitney's you're referring to are the M-1841's contracted and manufactured by Eli Whitney and Co. These rifles were
issued in great numbers to Southern State units.
Hope this information helps