CC Hunter, you and I have been following the same path in our research on Bill_S's casing.

I'll add a few other points of information, for any of the forum's readers who are interested.
The sole manufacturer of the .44 Henry Repeating Rifle was the New Haven (Connecticut) Arms Company. Its primary stockholder and director, Oliver Winchester, reorganized the company in 1866 and changed its name to the Winchester Repeating Arms Company. Also in 1866, that company began producing its improved version of the Henry, calling its new product the Model 1866 Winchester .44 Repeating Rifle. It was originally chambered to use the existing (civil war era) .44 Henry cartridge ...whose casing-length was .85"-to-.89", according to various info-sources.
Apparently because the "H" headstamp marking on those cartridges indicated "quality" ammunition to its customers, Winchester continued to use the H headstamp on its
rimfire cartridges for many decades ...even on other calibers of rimfire cartridges. For example, there is a slightly-postwar Spencer .56 cartridge which has the "raised" H headstamp.
Sometime around 1880, Winchester switched from the raised-H mark to a simpler "depressed" H mark.
Now, about Bill_S's mysterious 1.25"-long
rimfire casing with a raised-H mark-in-a-depressed-circle and "dual firing-pin" impression:
My research has turned up only three
19th-Century firearms which had a dual firing-pin. They are the .44 Henry Rifle, the Model 1866 Winchester .44 Rifle/Carbine, and the Swiss Vetterli 10.4mm Repeating Rifle (known in the US as a ".41 Swiss").
By the way... the Winchester .44 Model-1873 Rifle/Carbine used the .44-40 cartridge, which was strictly a centerfire, so that can't be Bill's casing. Also, the Winchester Model-1873 used a typical single firing-pin, so Bill's casing wasn't fired in that weapon.
Bill's "long" casing could not have been loading into a .44 Henry Rifle, so we can exclude that firearm as a possibility for Bill's casing. The .41 Vetterli's casing was specified to be 38mm (1.5-inch) long, so I think Bill's 1.25"-long casing would not have been used in a Vetterli.
Having eliminated the other possibilities, it seems Bill's 1.25"-long casing was fired in a Winchester Model-1866.
If I recall correctly, the US National Park Service's archeological report on casings excavated from the Custer battlefield mentions 'Short" and "Long" .44 casings with dual firing pin marks. Unfortunately, that report does not show "sideview" photos of those casings (just baseview photos). As you know, I don't mind sharing my research, so if you'd like to view that very extensive report and its photos (with lots of other Custer Battlefield cartridge and firing-pin info), go here, to see the 4.08-megabyte .pdf document:
http://www.nps.gov/mwac/publications/pdf/tech94b.pdf
At the moment, as we wait for more photos and precise measurement of the casing's "head" from Bill_S, I can only guess that the casing is a "long" .44 made by Winchester for use in the Model-1866 Winchester Repeater.