The proper way to refer to a singular gender-neutral person is in the masculine tense: "he", "him" or "his"
Granted, grammar rules are not fixed and I've noticed in the last 20 years that the general population is starting to use the plural gender-neutral person: "they", "their" or "them" as a singular gender-neutral person.
One of my favorite quotes is: "The dictionary does not define a language, on the contrary, a language defines the dictionary."
Because of this, it is just a matter of time before grammar books start referring to the singular gender-neutral person as "they", "their" or "them" instead of "he", "him" or "his".