Eowyn as commentary on gender roles is actually very interesting. She comes from a culture that values prowess in arms and glory in battle above all else, and also restricts war, the thing that grants both, to men.
Her going into battle is a defiance of that restriction, and a cosmically ordained one at that. She, a woman, was destined to be there. Illuvatar had a direct hand in her life.
Her choice to turn from glory in battle and to seek glory as a healer, considered more feminine in this mythos, is a rejection of the patriarchal import her people place on glory through the male province of battle.
What Eowyn Shieldmaiden of Rohan fears most is a cage, and she smashes that cage to bits.