"Anybody can dance" or "Everybody can dance", which is correct? Or do they have same meaning?
They can mean the same thing, but in some contexts only one of them is possible or they have different implications.
Both can mean that any person has enough skill to dance, but the connotations are different. Anybody can have a slight negative connotation: it can imply that dancing is so easy that someone who cannot dance is extremely clumsy, or that dancing is a worthless skill. There is usually no such negative connotation with everybody.
Anybody can dance. Singing takes skill.
Anybody can dance. But few can dance the tango.
Everybody can dance. It's a simple way of enjoying yourself.
Everybody insists that this is a rule with no exception.
Everybody can dance. Even if you're bound to a wheelchair, you can still move in an artistic way.
Anybody insists that any person who wishes to do it can do it.
Anybody can dance. The floor is open to everyone.
If it matters that many people are dancing at the same time, then everybody must be used, since it can be semantically plural.
The dance floor is huge. Everybody [who is invited to the party] can dance at the same time.