I'm a firm advocate of free speech. I think using any kind of power to silence any viewpoint is a dangerous business to start. I've never encountered anything so awful, dangerous, and evil that it can't be talked about.
When I try to argue with people that free speech is important, the argument I often hear back justifying people taking action to shut down someone they disagree with is "The first amendment only protects you from the government. It doesn't protect you from other people." This is a legally true statement. You have every right to protest and demand someone's business be destroyed or their employer fire them. However, I find that abhorrent to the philosophical reasons supporting freedom of speech.
A marketplace of ideas only works if people are not afraid of retribution for stating their belief. And I don't think where the retribution comes from should make a difference. In the past, the government has been the one shutting down free speech. It often still does. But more and more we're seeing people banding together and using new media to try to exert power or pressure institutions with power to use that power to silence speech.
Suppose you have a scientist who wants to publish a paper detailing how the dumping of certain chemicals contaminates water supplies and causes cancer. Now suppose there's a company that dumps a lot of these chemicals that's also a big donor to the private university that employs that scientist. The company tells the university they wont donate any more money if any faculty members publish any papers about those chemicals. The university then tells the scientist that if he publishes that paper, he'll be fired or never get tenure.
In this case you've had a private actor (the company) pressuring another private actor (the university) to silence the scientist and so hold back what could have been beneficial information to society. This is the danger of silencing speech whether it's a government silencing it or a private actor.