Hmm... "can't yell fire" is pretty much nonsense, but "private company" very much isn't. The rules for what the government can do and what private entities can are totally different, on purpose.
The problem is that this allows a neat loophole in which governments can sidestep any inconvenient civil rights obligations by privatisation of otherwise public space. For a real-life example, see https://en.wikipedia.org/wiki/Paternoster_Square#Occupy_Lond....