Don't know about ezmouse, but the Amiga supports 3 buttons out of the box.. So if that adapter is in the least bit sensible, it should direct the 3rd button to the right pin and it'll work out of the box.
Try it with MCP or some other util that lets you sample keypresses for shortcuts. You should get a middlebutton event when you click it.
At least my mroocheck works just fine with a three button serial mouse.
