While everybody seems like a plural noun, since it refers to a crowd of people, it is actually a singular subject. The key word to look for in "everybody" is "every" or "each," which implies "each single ONE." Everybody, nobody, anybody, somebody, each one, everyone, either, neither, and no one are all singular subjects, requiring singular verbs and pronouns.
Everybody loves Raymond.
Either soup or salad comes with your burger. [NOT: Either soup or salad come with your burger.]
Nobody wants to play the fool.
Each of those apples is bruised. [NOT: Each of those apples are bruised.]
Neither Casey nor Oscar knows where the office keys are located. [NOT: Neither Casey nor Oscar know where the office keys are located.]
Everyone here has lost at least one of his/her parents during the war. [NOT: Everyone here has lost at least on of their parents during the war.]
Note: When a non-gender-specific noun such as everyone, anyone, everybody, nobody, etc. is used, the subsequent pronoun must correspondingly be non-gender-specific as well. A common tendency is to use the word "their" as an alternative to the awkward "his/her." While this does allow for better flow or readability, this is grammatically incorrect. To use "their" as a valid alternative, change the subject of the sentence to a plural subject.
INCORRECT: Everyone here has lost at least one of his/her parents during the war.
CORRECT: The attendees here have lost at least one of their parents during the war.