The use of gender neutral pronouns in common English parlance is a practice which I strongly encourage. There are many times when the imposition of gender onto second personal pronouns causes much harm. I certainly have been the subject of much psychological pain due to the existence in English of gendered pronouns. My personal preference is to return to using e, which was common before the development of she to complement he.
Here follows my style guide:
He or She becomes E or Ey
His or Hers becomes Er or Eir
Him or Her becomes Em
As you may notice, these forms are derived by removing the th from they, their, and them. This emphasizes English speakers preference for the e sound over alternates like zie/sie/hy, and is shorter and simpler than others’ alternates.