Ζω (not ζώω ;) ) has the meaning of "I live, I am alive" and μένω means "I stay". So, αναπνέει άρα ζει=he breaths so he lives, but θα μείνω εδώ για λίγο=I will stay here for a while. But in this context above, both can be used. "I live in Greece"="Ζω/Μένω στην Ελλάδα".
Thanks a lot. And thanks for correcting me on the spelling. I will edit my question.