Everybody, even though it sounds plural, is considered a singular subject, so it requires a singular verb… HAS
Many people will say that using he possessive adjective THEIR is incorrect, because it is a plural and Everybody is singular.
To be formally correct you could say "Everybody has his or her own problems." But this can sound a little awkward, and it's now very common for people to use the 3rd person plural possessive as a gender-neutral 2nd person singular.
So it's generally okay to say "Everybody has their own problems" unless you are writing in a very formal context