Yes. All of these words are singular nouns, so you need a singular verb.
Nobody has arrived.
Does anyone know the answer?
Everybody understands.
That's the simple rule--> You can never use a plural verb form with any of these words.
Here's the slightly more complicated bit. Even when we use 'they' as a pronoun, the words everybody, anyone and so on remain singular. Here are some examples:
Nobody knows the answer, do they?
Everyone has their books.
Has everyone finished? Yes, they have.