> A private company could do the same thing.

Probably not, no private company has the ability to tell all its suppliers "thou shalt not do business with <some other company>". They lack the pull to compel them to do so. The closest would be things like what MS did back in the 90s, and got them in hot water with the anti-trust case against them.

Private companies are not bound by the first amendment, the fifth amendment, the administrative procedure act, etc.