The PC is a wonderful route so you should have a great time!
In regarding the wheels, I guess you could do it on 24 spokes but I would be careful about stomping on the pedals for having strong lateral forces on the wheels. The rim is also a big factor in the wheel strength and I don't know anything about your rim.
The thing to remember is that the more spokes you have on a properly built wheel, 1) the less chance of breaking a spoke since each spoke is carrying less of the load; 2) if one spoke does break, the rims is less likely to go out of true; 3) and the more stress/weight you can put on a wheel without issues. The inverse is true also with less spokes. So if you end up with a broken spoke, I would be very careful to fix it as soon as possible so others don't start to break or you could "taco" the rim which would put an immediate damper on the tour.
Another way to help strength the wheel is use as big a volume tire as possible. Of course, this heavier tire will cause more energy to be used but if it helps the wheel it might be a less expensive way to look at it to give your a buffer.
In the old days, a "bullet proof" wheel always meant 48 spokes. With today's rims, you can achieve a mostly bullet proof with 36 (touring tandems might need more). And for what you will spend on a decent set of new wheels, you might just look at a used touring bike with decent wheels already on it. I occasionally see a deal where the parts are definitely worth more than the sum.
Finally, take a look at the way Pete packs and seriously consider each item you carry if you really need it. The less weight you carry, the less likely you need stronger wheels.
Whatever you do, have a wonderful trip!