Just curious—since the probability of finding a hash that meets the target difficulty with enough trailing zeros is the same as finding a hash with trailing ones, why don’t we consider passing them through a simple NOT gate?...
I want quickly point out that basing the target on the number of 0-bits was discussed in the white paper, that is not how Bitcoin was implemented. Instead of comparing the number of zeroes, the hash and the target are both considered as numbers and the hash must be less than or equal to the target.
Counting zeroes (or ones) is not feasible because the difficulty can only ever change by a power of two. Your negative coin will have to be implemented differently in order to work.