No. Once that minion is killed, Eowyn's skirmish ends immediately.
Just so we are absolutely clear, in your scenario, there is one minion skirmishing Eowyn with a single wound on him, right? So when you activate Eowyn's text, the Shadow player has to place 1 wound wherever he likes. If he ends up wounding Eowyn's minion, that means the next time you activate Eowyn's text, the Shadow player will then have to place 2 wounds.