As far as I know, you are not allowed to use alternate verification methods. Or, rather, you are allowed to allow them but there must still be a checker that a user can use that will give the correct result.
As long as you express your route as logging caches in each of the specified counties, I think this should be a publishable challenge. I *think* you could require a specific order of the logs (from north to south, or whatever). Do note, however, that you *can't* require the logs to be in any specific time frame (in the same day, for instance).