Yes it is possible to get worms and other illnesses from biting and eating your fingernails especially if your fingernails are dirty and you haven't washed your hands or underneath your fingernails.
The way you get worms from eating your fingernails or biting your fingernails is not because of the fingernails themselves but instead it's because of the tiny little eggs and parasites that can hide on your fingernails and under your fingernails that you cannot see without the use of a microscope.
So if you bite your fingernails or eat your fingernails those worm eggs or parasites can get into your body which leads to possible illness and worms.
It's rare that people get worms from eating their fingernails but it is entirely possible and biting your fingernails is not good for your teeth either so I would avoid eating the fingernails or biting on them to protect both your health and your teeth.