Compare:
Everybody likes his or her own child. vs Everybody likes their own child.
Are both of these sentences natural to a native speaker? I want to mean the same thing in both of them. Can I interchangeably use them or should I stick with the second sentence?
TL;DR: They are both correct and acceptable. In formal writing, consult a style guide. In the non-binary community, "his or her" may be seen as exclusionary, but this has not yet become a standard rule, and some style guides still dictate it. In informal writing and speech, carefully consider your audience or default to "their."
The choice of pronoun in this case is a vexing one, that English has struggled with for generations. There is no single, universally accepted pronoun to use in this case. No matter which pronoun you choose, you will likely annoy somebody. The two examples you quoted will offend the fewest people, on average. Walking through our choices:
Everybody likes his own child.
Historically, it was common to simply gender an unknown person as male. This is now considered unacceptable by most authorities, but quite a few people grew up with this rule, and were told to avoid all other forms. As a result, you may see people insist that this is the "correct" pronoun. However, they are a minority in most circles.
Everybody likes her own child.
In modern writing, a feminine default is not uncommon. It is also not uncommon to see modern works alternate between female and male (when multiple unknown people need to be introduced). This also has the advantage of providing separate pronouns that can be used to distinguish between different people. This is mostly uncontroversial when used with specific hypothetical people, but more unusual when used with indefinites like "everybody" or "somebody." Also, the late Justice Scalia might make fun of you.
Everybody likes his or her own child.
Everybody likes his/her own child.
As "his" was rejected by native speakers, "his or her" became more popular in formal writing. "They" was seen as incorrect, or at least informal, by many writers. As a result, "his or her" carries a legalistic air. It's the sort of thing you might find in a contract or other legal document, especially in the "his/her" form. As I mentioned above, non-binary folks often don't like this usage. It's also pretty verbose, and is rarely heard in spoken English.
(Since I mentioned the non-binary community, I should also point out that people in that community have a wide range of pronoun preferences. When discussing a specific person, you should ask their preference, and respect it.)
Everybody likes their own child.
Singular "they" is one of the most controversial features of modern English. It is attested to at least the 14th century, yet is still seen as "wrong" or informal by many people. A lot of these folks originally learned to use the masculine default (as I discussed above), and were specifically told to avoid singular "they." Nevertheless, singular "they" is easily the least problematic option for informal writing and speech, and is increasingly accepted in formal writing.
One likes one's own child.
Finally, we can use "one" to refer to a hypothetical person, but it can't be combined with indefinites at all. This usage is correct and almost nobody finds it controversial, but it usually comes across as stilted and formal. You will probably never hear this usage in spoken English, and only rarely in written English.
Yes, they're both grammatically correct. If you're writing official documents for a large company or an academic paper for submission to a journal, they might have a style guide that would dictate the usage of one or the other, but in general, they both would sound fine to an English speaker.
