There may be more than one issue going on here, because I see the negative reactions to criticism as possible perfectionism, anxiety, or other insecurity. Any of those would be separate from quirkiness, nor related to narcissism.
In our family we can talk about being perceived as "weird" very easily, but that's because we've nurtured an environment where we can all laugh at ourselves and each other. Our house is a very silly place. We've embraced the idea that the whole family is weird, so it's easy to talk about how, as individuals, we exhibit that.
We also have a lot of serious talks about how we see others, how they see us, etc. I have a lot of conversations with DD where we hold up a social situation and explore it from all the different perspectives of the people involved. Some are from real life, and some are from books or movies. The goal is for her to acquire that ability to step back from her own experiences and see things from other lenses.
