Nice problem. I think they can only win if N=2
For N=2 the strategy is trivial, one always picks black, the other picks white
For larger N, notice that everyone's strategy depends on what they pick after seeing N-1 numbers. This gives N many N-1 uniform hypergraphs on the natural numbers and that encodes their entire strategy. This can be further reduced to coloring the N-1 subsets of the naturals by 2N many colors. From Ramsey's theorem there is a monochromatic N subset where all the N-1 subsets are coloured the same (from this compressed 2N color set). But then the prisoners can't escape, no matter what order we give those N numbers to the N prisoner they pick the same black white hats