Yes. Any group homomorphism fixes zero. Now, look at the set of things greater than zero. If the homomorphism is order preserving, then it must take the least thing there to the least thing in the image. However, as the image is surjective, it must then be fixed. Then, inductively, everything larger than zero is fixed. A similar argument works for things less than zero. In fact, this proof appears to merely require a totally ordered group, and not the lexicographic ordering, so this should work for any monomial ordering (that is, total order on Z^n.
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