"Yet one stood there still: Dernhelm the young, faithful beyond fear; and he wept, for he had loved his lord as a father." [The Battle of the Pelennor Fields]
A lot of people seem to think Eowyn was nothing but selfish when she left her people without protection to go to Battle. But did she leave her people? Or did she know she was needed in Battle more than sitting at home, waiting for something to happen?
For me, there are two kinds of people in the world. Those who wonder what it would be like to do something, and those who go and do it. Even though Eowyn rode to Battle seeking death and hopeless, I believe she had a last desire to save her people (the people who were fighting and aiding help, not those who remained at Rohan). For me, she went to the Battle of the Pelennor Fields not only to seek death and honor for herself, but to fight by her people's side. If it wasn't for Eowyn being there, nobody else could've killed the Witch King, because of the Prophecy.
"Begone, foul dwimmerlaik, lord of carrion! Leave the dead in peace!" [ Eowyn -The Battle of the Pelennor Fields]
Her first desire on riding to battle could've been to seek death and nothing else, but I believe after she saw her uncle Theoden dead, all she wanted to do was live. What good could she have done away from Battle? Her people had doubted her skills all of her life, yet for them she risked her life and fought an evil being thought to be immortal. I think that is being not only corageous, but selfless.
"For she is a fair maiden, fairest lady of a house of queens. And yet I know not how I should speak of her. When I first looked on her and perceived her unhappiness, it seemed to me that I saw a white flower standing straight and proud, shapely as a lily, and yet knew that it was hard, as if wrought by elf-wrights out of steel. Or was it, maybe, a frost that had turned its sap to ice, and so it stood, bitter-sweet, still fair to see, but stricken, soon to fall and die?" [Aragorn - The Houses of Healing]
Eowyn stayed by her people's side all of her life. She did nothing but protect them as she was allowed to, and not once she left them in danger. Because of that, she lived unhappily and frozen inside. When she saw the opportunity to actually fight, even though she knew she could die, she took it.
"...you stand between me and my lord and kin." [Eowyn - Battle of Pelennor Fields]
Not only for herself, but for Rohan.
"To health? It may be so. At least while there is an empty saddle of some fallen Rider that I can fill, and there are deeds to do. But to hope? I do not know." [ Eowyn - The Steward and the King ]
She was ill and sad after battle, yet she was willing to help in the Houses of Healing, help her people.
Eowyn was a heroine, not a selfish person. If it wasn't for her, the history of Middle-Earth would be a lot different. So yes, she disobeyed her uncle's orders.
"Well behaved women rarely make history." - Laurel Thatcher Ulrich
She was faithful to her people. Faithful beyond fear.