Everyone and everybody can be used the same meaning, you can use everyone instead of everybody. There are some people who use everyone when they write something and everybody when they talk with other. The reason why someone think everyone is more formal than everybody.
Then, everyone is one word, but every one is sometimes one word and somtimes two words. When you use everyone, it can use with singular verb. And when you use every one, every is adverb, so it usually use with plural verb.
Grammar Girl says 'Everyone are happy' is wrong because it's pairing the singular noun everyone with a plural berb, are. It means everyone should be used a singular noun. Then, I think everybody also is a singular noun.
Everyone and everybody can be used a singular noun, but the meaning is a plural. "Everyone is happy" in this sentence, some or many people are happy, not one person is happy. So using he or she instead of everyone is wrong. In my opinion, "everyone is putting a smile on their face." is right.
In Britain, it's standard to use everyone and everybody with a singular verb and plural pronoun. I think it is not a big deal about everyone and everybody is a singular pronoun because many grammarians decide to make sure yet.