AmericanMercenary says in Comments that the patch I found in my copy of H.G. Wells' War of the Worlds and The Invisible Man sounds a lot like the 4th Infantry Division's unit patch. I looked it up, and here it is--I guess I have the subdued version, shown here, because the others are bright green. Makes me wonder what the patch was doing there!
Needless to say, not having served, I won't be wearing this. But it's always interesting what you happen across at a yard sale (well, to me, at least) and thanks to my readers I now know what I actually found. Pretty cool.