This subject does not appear to have been addressed, but it seems that the simplest way to use the button with Equivalence Scoring is to pass the button as a prisoner rather than take it. Since taking the button is basically a non-game-ending pass worth half a point, passing the button rather than a full prisoner amounts to the same half-point gain. If the button is the size of a stone, it can also be used in Japanese counting by placing it on an intersection of territory that would have otherwise been occupied by a normal prisoner, thus representing half a point on the board. The remaining rules of Equivalence Scoring would be followed normally.

I noticed a few other half-pass related ideas, but this seems by far the simplest. If this analysis is correct, perhaps it should be mentioned on the main Button Go page or a related page.