Well, I had a little time today so I started hacking away at a cardboard box, found a bag of balloons (turned out to be very very small ones) and tried it out.
What I found was that the plastic BB just doesn't have the kinetic energy to push the cardboard flap hard enough to puncture the balloon. The weight of the BB is only around 12-15 grams, so even though it hits the cardboard with enough force to dent it, there just isn't enough mass to follow through. The cardboard flap absorbs the energy and, although it does push the flap, it is just not enough to push the pin into the balloon.
I have read somewhere that you can get BB's up to 20 or more grams. I don't know if that will be enough to do the job. On a more positive note, it doesn't take much more energy to do the trick. And when the balloon bursts it does send the flour "smoke" flying out all the box cutouts - looks pretty darn good!. I will try resting the pin on the balloon in a test tomorrow, I think that way the energy will go directly to the balloon without the "travel" shown in the original design.

I'm also thinking that it would not be that hard to make a "strike plate" switch which could be used to electrically actuate the effect. I'm thinking this one through now and will post any progress as I can. If anyone else is trying this please post your findings.