Say I have 13 endurence and my opponent has 14 will power on the field. Also suppose that he is at -14 Endurence. Come the recovery phase, what happens? Do we both lose?

nope, the person with Sinestro wins. Sinestros ability checks at the begining of the recovery phase. normal END check happens in the recovery wrap-up (after both players pass on an open chain, so after Sinestro's ablity resolves and wins the game for that player).

The game's normal endurance check comes as part of the recovery phase's wrap-up, which can't happen until the chain is empty and both players pass. Assuming your opponent has a non-stunned Sinestro going into the recovery phase, the effect will make you lose, leaving him as the winner.

Sinestro's effect triggers at the start of the recovery phase. When it resolves, your opponent has already won the game.