the cost of hiring replacements is unlikely to be unconsidered (aka "on a whim")
Perhaps not. But I was not the one who characterized their prerogative to fire, which I assume you presumed meant under any circumstances or reason, as a "rule of whim." Needless to say, incentives and disincentives influence decisions. Regardless of how this incentivizes or disincentivizes the party concerned, the prerogative still remains with them. So, for example, just as I would not object to an employer firing an employee for posting pictures of themselves using a competitor's product, I would not object to an employee quitting because their employer wears blue dress shirts at home.
So if we were to once again analyze your reference:
some corporations will fire you for posting pictures of yourself using a competitor's product
this is not a joke
Would an employer's firing an employee for posting pictures of themselves using a competitors product undermine, contradict, rebut, or refute my statement, here:
One only has "free speech" on one's own property.
No, not in the least. No more than it would for example if a girlfriend breaks up with a boyfriend after finding pictures of him locking lips with another girl on his couch in his apartment. The dissolution of an association, even unilateral and particularly absent of duress, doesn't qualify the prerogative of which I spoke. That is not to say, decisions albeit expressed through one's prerogative cannot be influenced by incentive.