To answer the question in the title:
Yes, if the rooster passes the blue egg gene to all of his daughters, then all of them could reasonably be called Easter Eggers.
Some Easter Eggers have two copies of the blue egg gene, some have one, some have none. So you will only know for sure if you test-mate him (breed to hens that have no blue egg gene, raise a bunch of daughters, see what color they lay) or do a DNA test (which does now exist-- a big help to anyone trying to breed chickens that lay colored eggs!)
Some hatcheries have flocks of Easter Eggers that they breed to each other, and some of the flocks do seem to be pure for the blue egg gene (just like purebred blue-egg breeds are supposed to do.) And some "purebred" chickens that are supposed to have two copies of the blue egg gene actually don't. So what the chicken is called, or what it looks like, is not always useful for knowing how many of a rooster's daughters will lay colored eggs.
The way you are spelling "Americauana" is not the way the pure breed is spelled.
Most hatcheries that sell chicks with any mis-spelled form of that name are really selling Easter Eggers. So some people say "Americauna" (wrong spelling) is the same thing as an Easter Egger, because from most hatcheries it is.
The correct form of the breed name is Ameraucana, which does not have the letter i anywhere.
For many breeds, getting the spelling a little wrong is no big deal. But since hatcheries keep using wrong spellings of this one, some people get pretty fussy about it. And then it's hard to tell who is fussing about the spelling, and who is fussing about whether the breed really is different than Easter Eggers.