I think he would remain a snow. In the winds of winter from Season 6 Lord Manderly calls him the White Wolf. That is not in reference to ghost. Normally the Stark banner is a grey dire wolf on a white banner. Bastards that become head of their own hose have to reverse the covers like in the blackfrye rebellion. So Johns banner would be a white wolf on a grew banner. Then in the opening credits for season 7 the dire wolf on Winterfell is a white wolf.