The way I see it, it's about MovieArwen being separated from MovieAragorn and pining for him.
But you've got me as to how it fits with the Houses of Healing scene...Maybe they thought that Eowyn could also say such words, since Aragorn is also 'turning away from her'.
Anyway, I find it a bit funny that Philippa says in the Extended ROTK that placing Arwen's lament during Eowyn's healing scene was meant to create a connexion between the two women. Well, the connexion is already, I think, pretty obvious!
(Oh no, here I go again, dumbing it down!

)