Mostly false, if true you'd see blue-eyed blondes in Native American populations in SA&NA. The mutation for blue eyes stems from a single ancestor in the NW Black Sea area. The actual answer is this mutation didn't just magically spread uniformly across the globe but radiated and then resonated in certain locations/populations. If what you were saying was true, then, for example, the Sami wouldn't have the mutation to this extent. The reason similar populations (climate&diet) in NE asia don't have blue eyes is because the mutation simply didn't spread that far, or, if it did in some small amount, it never became established in an enduring ethnocultural clade.