BER is caused by not enough calcium getting to the tomato, pepper, whatever, while it is developing. It's possible the soil does not have enough calcium in it to start with. Most clay has plenty but sandy soils can be calcium deprived. I don't know what soils you are using in that box so I don't know if it has enough calcium to start with. Adding more won't hurt and could solve the problem.
If the soil is too low in pH the chemistry doesn't allow the calcium to be picked up and carried to the fruit. Adding a little lime to raised the pH might help, not just because it has calcium in it but because of the pH levels. It may take a while for the lime to react but I like Rich's idea.
The normal cause of BER is water. If the soil is too dry there is not enough water moving to the fruit in the plant's circulatory system to carry enough calcium, even if plenty of calcium is present in the soil. I can't remember exactly why but the soil being too wet can cause a problem too. Anyway trying to keep the soil evenly moist and not going from too much to too little is what you are aiming for. Mulch is usually a very good thing to help with that. But try to even out the watering in that box. A little too wet is better than too dry but too wet can cause other problems.
Over-fertilizing with nitrogen can interfere with the calcium uptake. I knew all these others before this morning but I found this one doing some research. Excess nitrogen messes up the chemistry in the soil which ties up the calcium so it's not in a form to be absorbed by the plant's roots.
I haven't seen it myself but supposedly you can buy a special spray so the plant can absorb calcium through the leaves. This is supposed to be a quick fix but I have no experience with it.
What variety of tomato is he growing? Some varieties are more susceptible than others. I often have problems with paste tomatoes when the others are fine. Those little yellow pear tomatoes were really bad for BER when I grew them. They split like crazy too.
As a general note, compost made in part with chicken manure is going to be rich in calcium, especially from laying hens. Their digestive systems do not absorb all the calcium they eat, a fair amount of calcium passes right on through and out the rear end. Another reason to keep chickens and another reason to make your own compost. And use hat compost on tomatoes, peppers, squash, and eggplants. All these can get BER.
