Re: "anyone" is singular
According to traditional grammar: "his". Similarly, it is generally accepted that the word "mankind" does not just refer to males. A related example is the French word for "they" -- "ils" for men, "elles" for women, or "ils" for a mixed group.
To avoid being "gender-specific", I would say "his or her". To me, that sounds less clumsy than "their", because the grammar is not broken. Some people would choose to say "her". That's also fine, as far as I am concerned.
Many people would say "they" in this context. I don't go around correcting them, and I am sure that it is passing into normal usage. I just find it sounds a little bit wrong.