What I was trying to imply was that it's not enough to build a square wave generator to sound like a game boy. Even if that generator in the game boy was a perfect square wave generator, which I'm not sure it is.

The sound created by that generator passes through a fairly complicated filter known as The game boy's speaker. To properly create a game boy sound, you need to find or take an impulse response of that speaker and convolve it with the output of your oscillator.