Actually, I think it would be easier for private companies to get away with this than the government. After all, the only thing our Bill of Rights applies to is government (and really mostly toward Federal at that, unless you look at something the the Tenth Amendment). (Notwithstanding that privacy isn't a right...)

Private companies can place whatever restrictions and requirements they wish upon people in a business relationship. If you don't like it, you're free to conduct business with someone else. (Subject, of course, to market forces making such choices imprudent for them.)