Anyone/everyone

If we mean that entry is free,

Should we say Everyone or Anyone can enter?

You should say 'Admission is free.'

To take it a step further, you could say this:

Admission is free for everyone.

Ok, really funny
But I mean if you want to say that Anyone or Everyone can enter, any single one, which is the finest choice?

The previous responses were not intended to be funny, Anon. Your original question was not particularly clear.

You can say "Anyone can enter." Anyone means basically the same thing as everyone in this case, but with the added indication that it doesn't matter who the person is -- there are no exceptions.

What did you mean by "entry is free"? The usual meaning of that sentence would be that there is no charge to enter. If you mean that anyone is allowed to enter, you could say "anyone can enter" or "_______ is open to the public" -- but don't use the word "free" unless you mean that there is no cost.

You could also use the word 'enter' in reference to participation in a contest, for example.

Anyone can enter the contest.
It doesn't cost anything to enter the contest.
Entering the contest (participation in the contest) is free.

Should we say Everyone or Anyone can enter?

The correct choice for the subject position in this kind of declarative sentence (often with can, could, might, or would) is with any-. It doesn't matter who. It doesn't matter what.

Anyone can do that.
Anything could happen.
Anybody would think that he was crazy.

OK, I found it funny...about Admission is free for everyone........Just let's go even further....It is funny, at least to me.
So

How about this...

"She believes anyone has a special gift."

Everyone sounds better here, but I can't figure out why.

Anyone?

Yes, everyone is much better.

Anyone sounds like any one person.

Everyone means all people.