The expression itself refers to doing a small bit of physical labor.

It may be used to speak to someone you is not used to doing even the simplest physical tasks, and esp. someone who ‘doesn’t like to get messed up’ doing so. The suggestion is that this person, who doesn’t do manual (literally ‘hand’!) labor, who “doesn’t like to get their hands dirty” (another idiom), can manage to have nicely-groomed hands, perhaps even a manicure. Of course, a little physical labor might spoil the manicure.

To such a person you might sarcastically say, ‘Well, you wouldn’t want to break a nail” or “I wouldn’t want you to (have to) break a nail!” e.g., when they’re trying to get OUT of doing something. In this case, it is sometimes used LESS literally to simply “doing some HARD work” even if it is not PHYSICALLY demanding or messy.

Similarly, you may criticize someone who tries to avoid hard work by saying “she doesn’t want to break a nail.”

“Careful you don’t break a nail!” may be sarcastic or a bit more softly as a friendly joke.

On the other hand, people may use it to make the point that something will be quite undemanding, so easy, “You won’t break a nail.” In other words, you don’t need to worry about it. In this use it is similar to the non-literal expression “(you can do it) without breaking a sweat” ( compare “no sweat!”)

(Not sure but perhaps the idiom was also influenced knowledge of the practice of ancient Chinese landlords. They would grow very long fingernails which prevented manual labor AND were a sign of status, that they were far removed from common day labor.)