I think Cloudfare made the wrong decision here, but for me the reasons it's damaging to free speech are deeper than "is free speech at the level of government or private organizations?"
There's a couple of ways of looking at this. One is to say Cloudfare is a private company, they were free to make a decision, they exercised that right, and now white nationalists have the right to choose to go to a different provider. Others have the right to do business or withhold business from Cloudfare in response.
Another, though, is to say that Cloudfare is now in a unique position--by the CEO's own admission--and has power over another person's speech as a result. It would be akin to a husband controlling a wife's contacts with others. Sure, the wife could leave, but that's not really a good argument for the husband's behavior being ok; someone is, similarly in the hands of the company somewhat unfairly.
Yet another way to look at it is this: when Cloudfare decides it can and will make content-based decisions, have they now implicitly argued that when they don't remove content, they implicitly support that content, in that it's not aversive enough to remove? Where do you draw the line with that? And if a company nominally accepts that responsibility, does that mean we, in exchange, should allow them to regulate other traffic?
One argument for net neutrality is that while it binds a corporation's hands, it also frees them of responsibility for things they might otherwise be liable for. This was the bargain with phone companies, after all, with common carrier status. No one blames the phone company for supporting white supremacists because they carried their phone calls, but nor do they worry about the phone company dropping their calls because the phone company disagrees with their political position.
My impression is that the CEO of Cloudfare is freaking out at the moment because he realizes he has now made Cloudfare implicitly responsible for the content on its systems, and has opened up an argument against net neutrality. He's essentially saying to the government "please come up with rules that absolve us for responsibility in this situation."
If Cloudfare had simply said "we don't drop clients because of the nature of the content" they would have had a very strong position. Now they've opened a can of worms and have called into question their complicity in the content they carry.
They can't have it both ways: by saying that white supremacist groups are too aversive for them, they have now implicitly said that everything else is not too aversive. This is a very undesirable route to be going in in terms of freedom of speech.
For what it's worth, I also oppose network companies removing ISIS recruitment videos, all other things being equal. Now, if a court decided that the content poster/creator was in violation of some ethical and legal code to such an extent that their right to distribute content should be restricted, that's one thing, but that would require actual due process in a court of law.