Installing new toilet. Does the wax ring have to fit snug around the hole in the underside of the toilet?
The wax ring should seat so that no waste material can get out from around the drain when the toilet is seated on it.  When in place, it should look like this:





Now you may have purchased a wax ring that did not have the gasket built in, like such:



If so, and you get the sense that leaks are likely to happen, you should go ahead and clean that out and get the flanged kind.

Also, yes, when you apply pressure, it will smoosh out some, to create a seal, but the seal is intended to be between the floor and the toilet - it will not fill in gaps in the drain itself - that's the purpose of the flange.

Hope that helps!