I'm a 40 year old female, and I hated having my toenails cut as a kid. It wasn't the toenail cutting as much as having my feet held. My feet were so sensitive that it was torture. The only way I can explain it~ imagine being held down and having someone tickle your feet for several hours. Now put several hours of aggressive tickling into each and every touch. That is what it felt like.
In my 20's I developed 'Morton's neuromas' in both of my feet. Sounds scary, but it's just a benign growth on the nerve. One was killed off with alcohol injections, the other was removed surgically. I have no idea if that was part of my foot sensitivity. I also had a b12 deficiency,which led to painful numbness and tingling in all of my limbs.
Lots of kids hate having their nails cut, and it doesn't mean something is necessarily wrong. But, what you describe seems as if it is worth seeing a doctor for (to me at least). My suggestion is to let him try just using a low grit nail file. It will take a long time, but he can start and stop as he needs. Removing the aspect of having his feet held/ touched might be all he needs. If the nail file is still painful (it was for me), make an appointment with his pediatrician or a foot doctor. When you file, you create vibration. Nerves are sensitive to vibration. The doctor can determine if he has allodynia (normal sensations are interpreted as pain- touch, clothing touching skin,etc). This can be from things such as a b12 deficiency.
I would also make a video of him getting his nails cut. Sometimes doctors don't truly understand the severity of symptoms that you are trying to explain. After nerve surgery on my elbow, the wind blowing on my surgery site would cause so much pain I would curl up in a ball and rocks back and forth until the sensation passed. I would teeter between wanting to vomit and faint. Sounds ridiculous and dramatic, but luckily my arm surgeon had a lot of experience with nerve pain post surgery and properly treated me. In the video, get a close up of his skin so the doc can see if the skin color changes.
Please understand I am not saying something is seriously wrong. I just know what it felt like for people to assume I was being dramatic instead of recognizing I had a problem. Kids are usually truthful, and if they are attention seeking, they won't put on a show every single time. He very well might just be traumatized by the act of nail cutting because it hurt once before and he is scared. Never hurts to make sure though. Best of luck.