What is the position on head covering for Jewish boys and unmarried girls in publically funded schools?

I am not sure what you mean. For starters, unmarried girls need not cover their heads at all – in school or out. As for boys, the widely accepted practice among Ashkenaz Jews is that all men and boys (from a reasonable age) cover their heads wherever they go. Unless the covering of the head will have a significant negative consequence (antisemitic issues or livelihood issues), there is no dispensation for not covering the head. If one feels that there is a significant issue with covering his head, he should ask a specific question of a halachic authority.

Rabbi Moshe Rosenstein