"himself" is the technically correct word for singular reflexive when the gender is unknown
The submitted post is interesting because a lot of us prefer they/them for the gender-neutral 3rd person singular pronoun[0], and it raises an interesting corner case. I had to think about it for a bit, but I think "themself" sounds most natural to me.
[0] Why default to "he" when gender is unknown? Is correct number but incorrect gender better than "they", which is incorrect number but correct gender? "They" is natural to almost all native English speakers and has been used in that capacity in writing since at least Chaucer.
edit: Hm, now I wish I hadn't downvoted. Sorry about that. I suppose it's helpful for non-native HN speakers to know the "correct" word for academic writing, as much as it irritates me that that is what's considered correct.
The above notwithstanding, i certainly don't disagree that informal English has morphed towards usage of 'they' as the general gender neutral pronoun in all cases, as I myself just had to go back and correct "they don't" to "he doesn't" in the paragraph above when referring to the writer, even though his gender is clear from his name.
However I personally have no issue with 'himself' being the correct pronoun in the singular gender neutral reflexive for formal English.
Really the only way to decide what is "correct" where language usage is concerned is to see how people actually speak/write, AKA a descriptive approach. Do a corpus study and then you'll have some basis to make a claim about what is and isn't standard usage.
This is tangentially related: the English language is a bit unusual in how flexible it is to change along with common usage. Other languages can have their usage dictated by the government; Germany, France, and Japan have all had noted language reforms that in some cases radically changed the language. That is not to say, however, that all their citizens automatically start following these reforms perfectly because lifelong habits are hard to break.
I know about the French Academy, for example. I would have assumed their influence didn't extend much beyond orthographic/spelling conventions.
Thanks to feminist grammarians, "herself" is also correct. Or else you'll have to check your privilege, or some equally horrible fate. Though this suggests that the default gender for unknown animate third person pronouns should remain consistent throughout your entire written work. The writer should probably just pick his own gender unless there is a compelling reason to do otherwise.
One such reason could be the conditional probability for the subject being male or female, depending on what you already know about him. Engineer? Him. Elementary schoolteacher? Her. Someone got a mammogram? Her. Someone got a Darwin award? Him. If the conditional probability is 50% either way, you just pick your own gender, as above.
Today, it is technically incorrect, because society today acknowledges that women are full, living, breathing human beings and may in fact be referred to as such in polite conversation.