> Why is it I lose my right to speak if I'm in a group?
No one has suggested that you should. People have suggested that the corporation acting as such, including through its paid agents should not have the same rights as natural persons, acting alone or collectively, but that various classes of corporations should have constraints appropriate to their class.
One of the many reasons offered for this is that corporations are creations of government granted the use of the special form and legal fiction of personhood adn subsidized with various benefits with the purpose of achieving public benefit.
> In that respect, what makes corporations different from unions or advocacy groups like Greenpeace?
Unions and advocacy groups quite often are corporations. "Unions" and "advocacy groups" are descriptions of functions, "corporation" is a description of legal structure.