The words everybody, and everyone are indefinite pronouns, and they are always singular. They should be used with his or her, but you could use "their" instead.
Example 1: Everybody has his or her own problems. Example 2: Everybody has their own problems. To be grammatically correct you should use the singular pronouns, his, her, he, she. "Their" is more likely to be used, but is actually incorrect. Some other indefinate pronouns: one, anyone, someone, each, anybody, somebody, nobody, neither, and either.