I provided a direct axiomatization of integers for instance under the MO question <a href="http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers/">Axiomatic definition...</a> by @Victor Makarov, in the post <a href="http://mathoverflow.net/questions/23193/axiomatic-definition-of-integers/134770#134770">Part 2: Cyclands and integers</a>. This axiomatization makes no direct reference to the natural numbers, or to any linear order.