The sentence that has correct pronoun-antecedent agreement is "Each girl needs to bring her sleeping bag."
The antecedent in that sentence is "each girl", which is female and singular, so the pronoun should be (and is) female and singular too, "her".
In the other sentences three sentences, the pronoun and the antecedents do not agree in number (boys-his; no one-their; neither girl-their.)