"Anybody can dance" or "Everybody can dance", which is correct? Or do they have same meaning?
While there is nothing wrong grammatically with either of your sentences, there are a couple of potential problems with interpretation that I can see.
Strictly speaking, the word can means 'is able to'. If you are saying that everybody is able to dance then I present myself as living evidence to the contrary. Your intended meaning is more likely to be 'Anybody may dance', meaning that anybody is permitted to dance. Having said that, can is a commonly used informal replacement for may.
As to anybody vs everybody, the distinction in your context is minor. Anybody means zero or more of a given population, whereas everybody means all of a given population. Saying 'anybody may dance' implies that dancing is not restricted to a select few (probably the intended meaning). Saying 'everybody may dance' means pretty much the same thing, perhaps with the added implication that it would not be a problem should everyone decide to get up and boogie simultaneously.
A case where the distinction is important might be as follows. A group of people hire a bus for a day trip. The lady behind the counter at the bus hire company might tell you 'Anybody can drive the bus', but if she said 'Everybody can drive the bus' you would probably raise an eyebrow.
A: Who should I dance with?
B1: Dance with anybody!
(Meaning don't be choosy -- one partner is as good as the next)
B2: Dance with everybody!
(Meaning leave nobody undanced with: you have a long night's dancing ahead of you)
By the end of the song, everybody in the stadium was dancing with Lenart Krone, but Lenart was dancing alone. [...] Anybody can dance. You don't have to be Red Adair; just close your eyes and let the music lead you.
Example of usage:
A: You need to talk to somebody.
A: Anybody. Just talk. You'll feel better.