Urgh I can't understand why people would ever ask something so personal. How rude and weird of them.
As to how to deal with it....it's hard because unlike if, for example, it was a self contained group e.g. friends or colleagues, where you could just say whatever you wanted and then hopefully not be asked again, here it sounds like there's a revolving door of ever changing insensitive idiots, so even if you gave the "perfect" answer to one, another could ask again half an hour later.
So it really depends on how you want to play it. If you just want to shut them down a "I'd really rather not talk about it," would be completely fine, nobody could consider it to be rude, and might - might make them think about asking invasive questions.
Similarly you could directly address their rudeness e.g. to "What's wrong with your face?" you could respond, "There's nothing wrong with me" and/or "that's not a very nice thing to say." Unfortunately "Nothing, what's wrong with yours?" probably wouldn't come under good customer service, even if it would be entirely deserved!
Otherwise if you wanted to "take back the power" a bit you could have some fun making up increasingly bizarre stories "oh I used to work in the circus as a fire breather and when the elephant got loose there was an accident" or, dead pan "fight with a dragon," or even just wind them up by going "what looks painful? what scars?" completely puzzled so they feel like idiots.
Alternatively just completely ignore the question. Most people find it too awkward to repeat themselves, and if challenged you have plausible deniability to just say you didn't hear it/were concentrating so missed it. Will also have the advantage of them feeling awkward at being ignored without you having to do or say anything.
All depends on what you want to do and how supportive your boss is!