It does appear to be legit, but without a photo of the proof marks that are on the underside of the barrel breeches it is hard to say for sure. W.W. Greener guns were stamped with Birmingham proofmarks, and Greener's own trademark, which is an elephant. I only say this, because W.W. Greener was one of the most ripped off gunmakers that the world has known. There were countless Belgian made knockoffs, which were stamped with his name. He spent a lifetime fighting patent infringement, and down right fraud.
From just the photos you posted, it does appear to be genuine, and actually a somewhat higher grade muzzleloading gun from Greener than is usually seen in the U.S. due to the platinum blowout plugs that are present on the breech plugs.
Assuming it is a genuine Greener, you should contact W.W. Greener:
http://www.wwgreener.com/index.html
The company is still in business, and still run by a direct descendent of W. and W.W. Greener. Graham Greener is a very friendly and extremely helpful in dating, and giving information about vintage Greener guns. I contacted him via email about a percussion double I have as well, and all he requested were a few photos of the locks, proofmarks, and buttstock. Once I sent those to him he quickly replied with information about my gun. An exact date (day and month) for early muzzleloading guns is not usually possible, since I think most of those order books are no longer in existence, but he should be able to provide a year and grade for your gun.
Best of luck to you, and let us know what you find out. And most of all, take care of and enjoy your gun. It looks like it is still in fairly good condition.