There's a lot of arguments under the claim that because you are being censored my a private company rather than the government that you don't have the right to complain about it.

After all, they're private businesses they can do whatever they want, at least I'm not being oppressed by the government!

First of all, there's actually a notable connection between many of these large companies and the US government (or other governments) so they idea that they hold no responsibility in regards to free speech ends up a bit weaker. They're operating under the grace of the government they're under, if the government wants to shut them down it's bad news for them, so they generally want to cooperate and comply with government orders. While they're not being directly operated by the government, they do have to keep them happy.

Secondly, there's pretty much no offline public meeting place left to share ideas, if you get blacklisted from the Internet you have significantly less ability to express your ideas or opinions to be debated, debunked, or proven. What do you have left? Shouting in the middle of Starbucks on a busy afternoon? You're banned on social media, you can't operate your own website without being blasted off of the Internet and even if you did have valid claims nobody would be able to hear them. As a result your ability to exercise your speech has been dramatically diminished, while people you disagree with retain all of their rights because they have "the right ideas".

It's not hard to agree that neo-nazi ideas are unacceptable, inflammatory, and against my ideals and the majority of others, but it's not only their rights that are at risk. If the tables were turned and our idea's of what's right and wrong, moral, and otherwise acceptable were in the minority of opinion people would be extremely concerned about the right of arbitrary companies being able to restrict our ideas.

Ultimately we hope the best ideas can win out, but that's not possible if no alternative ideas are allowed to form. While the current problems may be from a quite arguably minuscule amount of people that are pretty disagreeable today, there's no telling what the future will bring in regards to new ideas which may shake up current social norms. What you use to hurt people you disagree with today, may be used against you tomorrow.

Finally, you can just ignore people and things you don't like. It's not a hard thing to do without censoring someone/something so hard that they get a platform under the reasoning that they're being censored by evil companies or governments. There was nothing forcing me to go to a website that I didn't want to go to, the majority of the sites that got censored would just be ignored or forgotten by the majority of people.
Nobody would be talking about these neo-nazi sites outside of a small fringe group of people if it wasn't for efforts to erase them from the Internet for having "the wrong ideas", there's always going to be people looking to get attention and the only way to battle that is to not give them the attention they seek.

People have the right to say things that are incorrect, disgusting, immoral, or otherwise wrong. I have the right to ignore those people. Censorship is not ignoring them, it's acknowledging them to the point of an extreme which you then have to justify with some moral grandstanding about how you saved the world from a problem they weren't having.