"Everyone" is singular so "his" or "her" own personality.

Their own personality.

You can use either or. Everyone has their own personality is correct and is gender neutral. Everyone has his or her own personality is OK, but sounds less formal and if you are writing say an essay, I wouldn't personally recommend it.

You can use his/her or "their". No worries. I myself usually use "their" in my written essays due to the formality.

In the past, many people said "his". In speech today, you should probably say "their", but some people consider it less correct than "his or her" when written. "Her or his" is also just as accurate as "his or her", but you might get funny looks. There have been dozens of attempts made over hundreds of years to make reforms (as merlyn says, these are called gender-neutral pronouns). Just do not say "its".