Looked to me like the button turned the fan off and on and the fan is just your run of the mill 120mm lighted fan. They don't have separate wires for light and fan power.
Either way it's really stupid to put a fan on a switch. How do you know when you want it on? The motherboard knows and will throttle a fan to take care of the temperature, running it as slowly as it can at all times to get the job done.
If you are anti-Christmas tree, although I haven't seen a blue christmas tree since the 1980s
, unlighted fans are cheap. Get a 3-wire one, plug it into one of the motherboard fan controllers, then connect the stupid switch to an air raid siren or something useful.
