Just a few quick thoughts on the matter:
Personally,although I respected and admired Eowyn's valor in the battlefield,I disliked her choise because it was an outcome of a broken heart in the way of since she cannot get what she whants,then she shall get nothing.And then,some days after,she decides to become a housewife and healer.
I do not think that this choice has something to do that much with feminism as with a general change of era.Middle-Earth was stepping from a time of war,blood and death to a new age of re-creation,rebuilding and healing.So,this is a time for healers and peace-makers,more mild-tempered characters like Faramir and Arwen,not warriors.Even Aragorn,who is introduced as the ultimate warriorand is by that idealised for Eowyn, appears to embrasse the role of the healer with much more joy than that of the warlord.Even if Eowyn remained a glorious,proud and alone Shield Maiden,she might have probably ended like Erendis,regretting for her stubbornes,bittered and wanting to take a last taste of the life she lost.
And concerning the stay-behind-unpraised by Aragorn...
If we see this statement from a different point of view,doesn't it reffer to the Rangers of the North in a way too?Weren't they a short of left-behind defenders of the North,whose great deeds were not praised as those of an army with shinny armors and shields,but even worse than women,they were treated as scoundrels and inhonoured?
As for some who accusse him of being the sexist warrrior due to that quote,do they really think it?With a more carefull observation to the story of his rising in power,he was not only motivated by Arwen,but also helped,if not pushed.A banner and a claim of origins can be doubted,but when it comes to the prophesy of the Elfstone and the chosen by the people name,things afe way more secure.
Now seriously,do you think that after all these political moovements would that woman would accept to become a breeding and sewind machine?
Or that a man with Aragorn's wisdom would not make use of such potential?