What do you think of asking someone to stop clipping their nails at work?
Does that bother you, and why do you think they'll listen?
It's their job to comply with the people who are paying them.
Depending on where they're doing it at work, that's certainly an uncomfortable situation to be in. Just try to approach the situation more tactfully, and remember that they may not have time to clip their nails in the morning.
I'll make them defensive about it.
There's no need to escalate the situation, but calmly tell them it is unhygienic and should be done at home or in the bathroom. Tell them it makes you uncomfortable and to do it before work.