This is an ongoing awkwardness in the English language. It still sounds awkward to many of us to use the third person plural to mean a third person masculine-or-feminine singular. However, I think it will eventually be adopted, simply because saying “him or her”, “he or she”, “him- or herself” (I prefer to use the hyphen) is even more awkward.

Note that in the original sentence “him or her self” is incorrect (hard to notice because the line break occurs there). Herself is one word.