A legal entity recognized by the State is all a corporation is. Unless the State provides that "recognition", there is no corporation. If the State does provide that recognition, there is one - whether a "private party" is involved or not. Lots of legal entities have few or no rights. Corporations have whatever rights the State gives to the corporations it "recognizes". In the US, an activist Court early last century bestowed many of the rights of actual personhood and citizenship on US chartered corporations - a bizarre legal development, too widely imitated by the unwary and unwise.
There was certainly capitalism before that, and would be without it.
Actually, just about every person and legal entity has the right to not be robbed, defrauded, etc. Corporations are no different.
And while an absence of corporations, in and of itself, wouldn't prevent a nation from being considered "capitalist", I'd say that a ban on corporations would. The existence of corporations doesn't violate the rights of others, so a ban would be anti-capitalist in nature.