The plural their has been used with everybody/everyone for centuries and I for one consider it perfectly acceptable: Everybody takes off their hats. Gonna for going to is informal, slang, colloquial or nonstandard English - or whatever you prefer to call it. Anyway, it shouldn't be used in serious writing.
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AnonymousHow does one refer to the indef. pronoun Everybody in British English. Personally I think one should do so with a plural pronoun but I have heard that this would only be possible in very informal situations. Apparently the plural pronoun must be replaced by "his or her" (cf everybody must put his or her coat in the corner vs their coat).What