Cloudflare isn't preventing anyone from saying anything. All that is happening is Cloudflare is using it's freedom of association not to associate with this website. This has nothing to do with 8Chan's ability to publish what they want - only their ability to use Cloudflare's services to do so. As the article mentions - because the US has the principle of Free Speech enshrined in law, 8chan has the ability to go and use other services, or to develop the services that Cloudflare provided. This will not impact 8chan's ability to publish whatever they like.
Now there is a theoretical point of view, that if a company has a monopoly - it is effectively able to police speech, but that's absolutely not the case here, as is demonstrated in the past by companies going elsewhere to exercise their right.
What is being proposed as an understanding of Free Speech is not allowing private individuals from refusing to service to you. This would seem to be mandating someone to act - which is a principle quite far away from any law I've heard of.