I think it depends on which logic you want to follow. Do you want to explain why the people eat? Then you'd say "Azok esznek, akik éhesek." If you want to say instead what hungry people are doing, you can go for "Azok, akik éhesek, esznek." Compare with German: "Es essen die, die hungrig sind" vs. "Die, die hungrig sind, essen." Both are totally valid, the first version just sounds a little better.
I think "Those eat who are hungry" should be accepted, too, even if it sounds a little troubled. I'm not a big fan of subordinate clauses in the middle of the sentence anyway.