Corporations don’t have “all the rights of an individual.” You are making the common mistake of misunderstanding Citizens United. The majority wrote, "If the First Amendment has any force, it prohibits Congress from fining or jailing citizens, or associations of citizens, for simply engaging in political speech."
The key thing in Citizens United isn’t that “corporations are people,” but that an association of people be the same First Amendment rights as an individual person. And that makes sense: a newspaper is published by a group of people, spending money to distribute their message. A newspaper editorial is published by the paper, not by the individual. So restricting an association of people’s First Amendment rights, you would necessarily restrict the right of a newspaper to publish, unless the newspaper is completely self-funded by the writer/publisher.
So no, corporations don’t have all the rights of an individual, they can’t for example, vote in an election, join the military, hold elective office. A corporation is nothing more than a group of people sharing a common goal. A corporation is no different from a club, a union, a rock band, or an activist group. It’s just a legal structure that allows people to join together for some common purpose.