"There are actually two issues concerning this topic: Are the words everyone andeverybody singular or plural? And can I use a plural pronoun (such as their) to refer to these words? Grammarians actually agree that the words everyone andeverybody are singular. Grammar Girl (that is I!) herself explains the answer in her book. She says, everyone sounds like a lot of people, but in grammar land, everyone is a singular noun" (Grammar Girl).
"Everybody: Singular or Plural." Grammar Girl: Quick and Dirty Tips for Better Writing. 2008. http://grammar.quickanddirtytips.com/indefinite-pronouns.aspx
"Usage - Subject - Verb Agreement." 2010. Towson University.
http://www.towson.edu/ows/sub-verb.htm