There's so much Tolkien DOESN'T tell us about Eowyn's inner life, especially during her time in the Houses of Healing. We are only given brief vignettes, and this makes it seem like her feelings towards Aragorn and Faramir change on a dime. But I think Faramir makes a very important statement about her in "The Steward and the King" that isn't often discussed:
"Then if you will have it so, lady," he said: "you do not go [to the Field of Cormallen], because only your brother called for you, and to look on the Lord Aragorn, Elendil's heir, in his triumph would now bring you no joy. Or because I do not go, and you desire still to be near me. And maybe for both these reasons, and you yourself cannot choose between them. Eowyn, do you not love me, or will you not?"
A few paragraphs later, Tolkien states, "Then the heart of Eowyn changes,
or else at last she understood it. And suddenly her winter passed, and the sun shone on her."
Faramir is a very perceptive man! I think the implication is that Eowyn's feelings for Faramir have grown and blossomed over an extended period of time, during which she has been in a great deal of emotional turmoil over a number of other issues: dealing with the aftermath of her injuries (how helpless and useless she must have felt with BOTH arms injured -- one broken, the other cold and lifeless!) and the lingering effects of the Black Breath and Grima's dreadful whisperings; coming to terms with the deaths of her uncle and cousin (whom she really never had a chance to properly mourn); facing the possibility of losing her dear brother, the very last of her near kin; dealing with the guilt she MUST feel for having left her people leaderless; and looking day after day into the growing dark of Mordor and the virtual certainty of annihilation.
Plus, she clearly feels that she has ridden all this way to find glory and honor and
freedom in death, only to find that she has basically traded one cage for another! The image of her restlessly pacing the halls and gardens of the Houses of Healing -- unable to leave, unable to fight, unable to do ANYTHING -- is very palpable. "I cannot lie in sloth, idle,
caged."
The poor girl -- and she is really just a young girl at heart -- is utterly and completely devastated and confused! So, when beyond all hope the eucatastrophe occurs, and the world DOESN'T come to a dreadful end, and everyone DOESN'T die, and she must face life once again, she simply doesn't know what to do with herself, or all of the muddled, confused, unidentifiable things she is feeling.
It is Faramir, bless him!, who brings clarity and light and love to her at last. It isn't that she suddenly stops loving Aragorn and starts loving Faramir, but that she finally at last understands her own feelings.
It is also important that Faramir confesses his love to her first. She has already been through the terrible experience of revealing feelings to Aragorn that could not be returned. Part of the reason she cannot go to the Field of Cormallen is that she is too ashamed and embarrassed to face him again. In order for her to be able to admit to her feelings for Faramir, she needs to be certain of his love for her first.
Ultimately, Eowyn's greatest difficulty is that she is, as Gandalf pointed out, a person born with the spirit and courage of a man trapped in a maiden's body. On top of that, she has been raised much more like a young man than a young woman, and spent the vast majority of her life in a household completely dominated by men and constantly full of warriors coming and going. She has probably never had any suitors, other than the odious and unwanted attentions of Grima, and she really doesn't know how to BE a normal young woman of her time.
Merry brought up the question of whether there were other women who had been trained to fight like Eowyn -- other "shieldmaidens." Since the term shieldmaiden even exists, I would say yes, but they were probably not trained formally as Eowyn was. I suspect that many women of Rohan learned to handle weapons -- especially the women who lived out on the farms and pastures of the Westfold -- not with the sense of turning them into an organized fighting force, but so that they would have some skills to defend themselves in case they are attacked while the men are out in the fields or off on patrol. But Eowyn herself defined the place of women in Rohan when she said so bitterly to Aragorn, "All your words are but to say: you are a woman, and your part is in the house. But when the men have died in battle and honour, you have leave to be burned in the house, for the men will need it no more."
In Eowyn, Tolkien created a very unique character for her time and place -- the rare female warrior. She represents Brunnhilde and the Amazons -- all of those "unnatural" female warriors who must be tamed by the hand of Man. And Eowyn is, in a sense, "tamed," but in a far more civilized manner certainly than Brunnhilde -- not by a man who bests her and claims her in marriage, but by a man who declares his love, upholds her worth and valor in his eyes, stretches out his hand, and gives her the CHOICE and the FREEDOM to decide her own future.
And that's really what she wanted after all -- FREEDOM.