No. I think Michelle Obama was right. I used to think "follow local custom" but I've actually changed my mind.
Muhammad instructed his followers to dress certain ways (actually he might not have, this is open to debate) but I don't see any reason why non-Muslims should follow Islamic dress. Just like they are not going to pray five times a day.
Clearly it would have been inappropriate for her to wear a strappy vest or strapless dress and she was mindful of a level of expected modesty but headscarfs are particular to religious thinking and I don't think should be worn by people who don't subscribe to that religion or that interpretation of that religion.