I know this site has to be very conservative when discussing these kinds of issues, but let's be honest: if every teen who popped a blackhead with the fingernails got an infection, there wouldn't be a bottle of antibiotics left on the shelves. The best you're probably going to do is to get them to wash their hands first.
I've used the blackhead removal tool, known affectionately in our house as "the zit popper." They do work, if the blackhead is sufficiently "juicy." Yeah, sounds gross, but this isn't afternoon tea we're talking about here. But if the blackhead isn't in an area where the zit popper can get a good bead on it, and the blackhead is itching like crazy, well, fingernails become an option.
As much as doctors advise against it, popping an itchy blackhead is sweet relief. I had one on my forehead years ago, and it was itching so, I actually popped it in my sleep. True story.