I can't clobber you over the head with a product I bought just like how many countries have laws to say I can't kick you out of private property that's also a public square.
However, you can evict me from the property and charge me with trespassing, which is closer to whats happening here.. edit: Hell, you can even infringe my 2nd amendment rights and ban me from having a gun on your property!
Also, I'm not sure 'private' property can be a 'public' square. At least in New England, the town commons was public land ?
Quickly looking this up brings me to the supreme court ruling, Pruneyard Shopping Center v. Robins [1], which involved laws that were specific to the state of California.
Notably, the applicability of this ruling was narrowed over time. Despite this, it has been brought up as an argument in situations with circumstances that differed from those of the original ruling, where questions of its applicability can be easily raised.
[1] https://en.wikipedia.org/wiki/Pruneyard_Shopping_Center_v._R...
edit: I seem to remember my son's class not being able to petition in front of a grocery store in California like 20 years ago...Wonder if it had been narrowed by then?