I'm on my second button for this year. I know roughly when and where it disappeared (night fishing in ten inches of snow in State College in January), so there was little hope of finding it until things thawed out. Seeing as I didn't discover it was missing until a week after it dropped, it's probably gone. Last year, I made it the whole year with the blue button, but I recall bending the pin a little bit so that it stuck better. I was concerned that I was going to break the pin or the backing though and was trying to figure out how to get a sturdier pin in place to replace the flimsy one.
I like the button over a paper license for a number of reasons - with a flapping license holder, I inevitably end up getting my line tangled on it, whereas a button has a much lower surface footprint and remains flatter on my chest pack. In theory, it should also be harder to lose or get caught on something, but I've dashed that theory. You also don't have a rectangular yellow or white block exposed to the fish.