In particular, free speech is a right of "every ... association representing a segment of American life and taking an active part in our political campaigns and discussions" [1] -- unions, guilds, consumer groups, religious groups, newspapers, political parties, and so forth. Those groups are made up of individuals united for a purpose, and their choosing to unite should not penalize them by restricting their ability to speak. Justice Kennedy made a great comment to this effect: "wealthy individuals and unincorporated associations can spend unlimited amounts .... Yet [under the Austin decision] certain disfavored associations of citizens — those that have taken on the corporate form — are penalized for engaging in the same political speech .... When Government seeks to use its full power, including the criminal law, to command where a person may get his or her information or what distrusted source he or she may not hear, it uses censorship to control thought. This is unlawful." [2]
[0] Pembina Consolidated Silver Mining Co. v. Pennsylvania, 1886. Yes, that's eighteen eighty six.
[1] US vs UAW, 1957
[2] Citizens United v FEC, 2010 - page 40 of Justice Kennedy's majority opinion
The question is not whether a corporation has rights similar to a natural person's -- in order to work at all, a corporation must have some of those rights. For example, it must have the right to own property, the right to enter into contracts, the right to bring cases in court, and so on.
The question is which set of rights corporations should have, and to what extent they can be regulated. Which is far from easy to sort out or solve with simple black-and-white rules.