How do you deal with people who fish for compliments out of insecurity?

I have a few people in my life who will do this regularly: They strike up a conversation, and then, at some point, sprinkle in some negativity about themselves. Most recently, I had a teen try this with me. This is a girl who is not insecure in general, but very obviously was insecure about the topic at hand.

I called her out on it, gently, and we kept talking about the issue in a more honest way. I now wonder whether I was right to address her trying this. She was slightly embarrassed by me noticing, though I did tell her I thought it a very common thing.

What would you have done?