many moons ago I studied sound engineering and part of syllabus was studio design and soundproofing so here goes..
for one.. I don't think any box will be able to deaden the sonic boom unless it is totally sealed which is obviously not possible here.. but here goes..
try and find yourself a material called rockwool.. a large musical instrument place should be able to lead you in the right direction.. get rid of the foam in the side panels and fill it with the rock wool..
I also think the muzzle blast is going to destroy those egg cartons pretty quick.. something more robust and less ridgid and thin is going to be needed... perhaps a cheap alternative would be eggshell foam in the 50mm variety - available at Clark Rubber
Is the lid you are using insulated as well... because I feel if it is not, you are going to get some interesting reverberation in that box.. think "fully sick subwoofer"..
you will also need to cover all the ply in the box with your choice of soundproofing.. in this case egg cartons.. the dividers that are not covered are also going to reflect the sound quite a bit - again subwoofer box..
I reckon it will dampen the sound, however, you wil find that with the box not being sealed, all you will be doing is redirecting the sound .. any chance you can make those divider/baffles larger.. i.e. the holes in the middle of them smaller..
interesting project though.. I once helped a mate build a soundproof box for a very noisy generator.. sealed box, fans and baffled exhaust/intake ports.. for 2mt away you could not hear it
let us know if you need anymore advice or a head to bounce some ideas off..