You and eshan are correct, but there is another angle which you both missed. The modal verb 'will' serves several purposes, one of which is to indicate willingness or intention, another is to form the future tense. "I will not eat beetroot" could either be a refusal, or it could simply be a prediction about the future.
The past tense of 'will' is 'would' and, as such, it has these same two purposes in the past: it can describe an intention (for two years, the boy refused to remove the helmet) or a prediction about the future relative to the past time being described. The latter use is quite common: for example, "When I left home that morning, I feared that I would never see my father again". This example clearly doesn't relate to an intention, I loved him a lot!
Your sentence could be read in either way, but when I read it I interpreted it as 'future relative to the past', I think because I would have used an unambiguous and more emphatic wording if I had wanted to focus on the boy's intentions.