I found these articles about RFID chips, as well as several articles about magnetic implants and rejection. (warning: bloody pictures of home surgery in the last three articles)
Problem 1 - implantation:
The chips will obviously be made to last a lifetime, and will be small enough to be implanted through a thick needle when the children are as young as three years old. Would it be plausible to make chips that small, but functional with modern technology? And how big does it have to be so it is guaranteed to stay in the hand, with only minor migration after implantation?
Problem 2 - removal:
In my story, two siblings try to run away from this dictatorship, and need to cut their chips out. The girl is easy, she is helped by people with hi-tech equipment and gets her chip removed easly. But when her brother escapes, there are only the two of them and limited equipment. How can she destroy or remove a small ID chip?
She has experience with medical treatment with limited equipment, and has cut people open before. But he will not be drugged down, and they don't have a local anaestetic to use on him. I'm thinking tying his hand down to keep it as still as possible, and also cutting off the blood flow for a short period of time to reduce bleeding while she cuts.
ETA: The reason the ID chip is implanted in the hand is to be able to simply put it on a reader to pass through gates/identify themselves when using computers/enter buildings/buy things. That would be harder with a wandering chip.