I think that phrasing is deceptive. The question isn't whether corporations are people with rights, but whether people retain their rights when they coordinate the exercise thereof through a corporation, which includes non-profit groups. It should be a red flag that abolishing that right would abolish the free press without some special exemption. That requires the government to start defining who is press and who is not, which if anything is a quickly fading distinction. The distinction between MSNBC/Fox News and a SuperPAC is not so clear. (Thankfully, we have such an exemption because the first amendment particularly protects the freedom of the press, in addition to the general freedom of speech. There's a lot of examples of redundant provisions in the Constitution, and I think this is one of them. The freedom of the press flows naturally from all of the other freedoms in the Constitution.)
The particular case was actually about a political group that made a film criticizing Hillary back in 2008 during the primary season. The government wanted to ban them from promoting it during the two months leading up to the election. How can anyone think that's proper?
Let's say that you and I shared a passion for a political idea. We want to convince people of it and effect change through the political process. What would we do? We'd set up a non-profit (a corporation), convince donors of our cause, spend their money on pamphlets, broadcast ads, internet ads, discussion forums, etc. We'd support candidates that back our idea, and oppose those who oppose it. That is freedom at work and it's exactly what the Citizens United decision protects.