Originally the 76th was issues British Enfield rifled muskets purchased in England by the State of New York. As far as I have been able to discern they retained these guns until the end. However it is very likely that there was a smattering of other .577/.58 caliber weapons in the ranks, perhaps even a few 1842 muskets. Considering the losses the 76th took in combat replacements may have been issued any substitute standard arm.
As far as re-enacting the 76th we all carry Enfields, they were the original issue weapon and they are cheaper than the Springfield reproductions on the market. Additionally they have better sights and are blued so they are easier to maintain(they do not rust as readily as the bright finished Springfields). The blueing also makes for a better sight picture (less glare) which is helpful if you intend to hunt or target shoot with your musket as well. There are three brands of reproduction Enfields available today, Euroarms, Armi Sport, and Parker-Hale. Of the three the Parker-Hale is nicest and most accuratly reproduced it is also substantially more expensive than any of the other rifles, Springfields included. The Parker-Hale was made some years ago in England and today production has resumed there. Armi sprot rifles are nice and the cheapest of the three, they are better finished than they were in the past and they have correct barrel bands, Euroarms are slightly more money and slightly better made, unfortunately they do not have the right barrel bands and if you decide to de-farb your rifle they will need to be replaced with Armi Sport bands.
If you decide to get a Springfield or other musket you'll be fine and they do have some advantages over the Enfields, eg.1861s are lighter and have a better stock "feel" than Enfields.
Recently we have discovered that the 76th was equipped initially with Enfields and then by the close of the war they turned in Springfields. As of Gettysburg they apparently had a mixture of both.[This info was attained from ordnance returns and ammunition requisitions found in Record Groups 94,109, and 156 in the National Archives and Records Administration, Washington, D.C.]
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