I usually do the opposite. Sometimes when someone I manage asks me how things are going, I jokingly tell them I'm trying my best not to get fired. I find it helps break the ice, and shows them that other people are struggling with that same insecurities. Reading some of the comments on here I'm now concerned that such comments might also amplify a general latent fear of getting fired, so I'll probably stop doing this.
Yeah probably best to not make it about being fired—since your boss is ultimately their boss—makes them wonder when the axe might come for them next… That said I’m definitely a fan of some self-depreciating humor to help others relax and build rapport. Humor can be a wonderful tool really.