Not the 1st gen, but I installed a similar button on my 2022 BRZ. I got mine from Import Image Racing, but it looks exactly the same as the one you linked and came with the same hardware.
Here's a video showing how to install it on the 2nd gen, but the process should be the same for the 1st gen.
I'll add that I didn't follow the video exactly. I just left the switch assembly in the car while I removed the button with the wire tool. There's really no reason that you'd need to take the switch out of the car for this. In fact, it might even be easier this way since the switch is held firmly in place, which makes it easier to pull the button out. You also don't have to go through the hassle of taking the assembly out and putting it back in.