Every person has the right to free speech. But they can then be held accountable for that speech. Thus libel, slander, etc.
So congress introduced Common Carrier status, in which telecommunications companies could then be NOT held responsible for data that simply passes through their network.
Now let me get this straight, Verizon is trying to claim anything passing through their network is their free speech? This raises two problems for them.
First, if it is all their speech, then they can thus be held accountable for everything going through their network, as common carrier only applies to OTHERS speech going through them as a conduit. Meaning they can be held responsible for every libelous, slanderous, copyright-violating, child porn-downloading piece of data going through their network.
Second, this becomes straight out copyright violation. If I post something online, it is still copyright by me. Now Verizon is trying to claim it is THEIR free speech, not MINE. Essentially violating my original copyright by asserting their ownership of it because it happened to go through their network. It would be the equivalent of Barnes and Noble asserting copyright of any book on their shelves because it went through their store (by saying it is THEIR creation, not the original author's).
Both these arguments pretty much break down Verizon's free speech argument, without even delving into the 'corporations are people' argument. They would NEVER want either of these to be true, as it would open them up to massive amounts of civil and criminal charges. But if they are claiming that everything on their network is THEIR free speech, then one or both must be true, and they must then lose common carrier status.
And incidentally, they can't claim the whole private property rights either, because THEY are the ones letting people use the network, and THEY are connecting to peers specifically to allow the provider's content (youtube, microsoft.com, whatever) to get to the people who are paying them to use their network. You can't complain about people walking across your private property if you are charging them specifically TO walk across your private property. If they want to claim private property, they should then simply be not allowing people to access their network, or peering with other ISPs to allow traffic to flow through their network. Of course, that then means they have no customers and no business, but it would protect their 'private property.'