In the UK anyone under the age of 18 must wear a helmet when riding a horse.
I can see where the idea of the danger of breaking your neck comes from.
Way back when I was much younger and probably before any of you were a twinkle in your father's eye, riding hats had solid peaks to them and were either held on with an elastic strap or just by luck!
It was then brought in that chin straps were to be worn.
So, with a solid peak the danger of braking your neck was greater because of the danger of whiplash. I recall a child being seriously injured by catching her head on the forehead by a branch and being dragged off backwards, she broke her neck but had no ill effects once it healed.
Modern hats are much safer and without a peak there is no danger of anything catching under the peak. The peaks that are on some helmets are all collapsable.