I had the same problem drilling the button holes through 3/4" MDF - I thought I'd be smart and drill from the back, but when it went through the front, it split some of the MDF away from the spaces in between the holes.
I had nothing to lose, so I threw in some of this really thin spackle stuff, let it dry, sanded it, did it again, and it worked out very nicely. It was 2 days from first spackle to finished.
I was just too stubborn to give up on the CP board, even though it looked ugly.