You are being overly sensitive about this. Your disgust is not justified. Here is my outside perspective (as an in-office nail clipper myself):
As with all office space sounds you may find irksome, invest in some decent noise cancelling headphones and install a mirror above your PC monitor so you can see people enter your cubicle.
It's not your place to tell people what they can and cannot do within the office. You're free to have whatever expectations you wish, but keep in mind that not everyone shares those expectations and that yours are not more valuable than theirs.
Given that the soles of most shoes have a high likelihood of having fecal matter on them, fingernail clippings (keratin) do not appreciably make the floor any less clean. The floors are likely vacuumed nightly anyway.
My advice: grit your teeth and deal with it. People who are fastidious about their hygiene and grooming habits are not likely to change them because someone asks them to.
The carpet is likely vacuumed nightly. What impact do fingernail clippings in the carpet have on you? Have you experienced problems with this already?
It may be your opinion that grooming should be done at home, but other people have differing opinions. It would be the height of unprofessionalism to engage in these behaviors while in conversation with a co-worker or client, but that doesn't seem to be your complaint. Also, your blanket statement of "every other hygiene behavior" being performed at home is preposterous. So people shouldn't wash their hands after using the bathroom or wipe after defecation? Ridiculous!