Public servants, not individual citizens. The individual citizens are incidental and not important.

DHS is a public institution staffed by government agents acting as public servants, not as individual citizens.

OP specifically referenced this applying to both government agents and private entities.

OP only mentioned private entities conspiring with the government to violate the constitutional civil right to free speech. Context.

Are you suggesting that I as an individual lose rights if my employer starts working with the government?

Of course you do. You probably lost rights before then if you signed an employment contract, and your employer lost rights when they started working with the government. You lost rights the first time you registered with paypal or facebook (forced arbitration clauses.) You lost rights when you signed your lease.

I don't understand what's supposed to be shocking here. There's this phrasing that seems to indicate that this is an absurd situation rather than a commonplace one.

Yes, obviously. That's how it works.

It just seems weird to suggest that I can be like a random IT worker at some tech company and can post something like "All hate speech should be illegal" and that will be fine. Then my company signs some contract with the government and suddenly tweeting that again would result in prison time.

is "conspiring" somewhere between speech and action or is it firmly within one or the other ?

GP didn't use the word conspiring, I did. But to answer your question, conspiracy is an expression of intent. It can be merely speech but usually involves some action to back up the speech. Because, again, conspiracy is a question of establishing intent.

For example, three people discussing plans for a bank robbery are conspiring. Likewise, the same three people sharing only a knowing glance before splitting up and regrouping with masks, guns, a car and a map with the local bank circled have also conspired to rob a bank. One group only spoke and the other didn't speak, yet both conspired.