It is your business, but I personally would not do this. When wearing a business hat you need to respond to rudeness with polite firmness.
"fix it!" -- "please give more details" or "sorry, this feature is not a near term deliverable" (which is a polite request to get lost). "I am going to a competitor" -- "we think our product is better, but encourage users to try alternatives", etc. My 2c.
"Who cares?" Presumably, the person who need a steady stream of new clients to survive, many of whom do a bit of Googling that would uncover evidence of dickish behaviour (reviews, reddit posts, tweets, etc.), whether or not the recipient deserved it.
There's also the fact that being a dick to rude people doesn't feel good to everyone. I can only truly speak for myself here, but being rude in proportion to some random person who started it and totally deserves it leaves me feeling bad afterward, like I allowed myself to get trolled. An angry client email, a profane reply to an innocuous online comment... You have the luxury of asynchronous communication to take a moment, let the blind rage pass, and reply in the even-keeled diction of business communications, maybe even killing them with kindness. Or just ignore 'em.
Do not work like that. When I read reviews, I quickly notice if the reviewer seems to have an axe to grind. Reviews of true bad service seem to have a different favour than the reviews given by the people who are just trying to bad mouth someone/company, they too often try to make themselves look like angels and how dare this big bad person not meet their expectations.
Regardless of how entitled and axe-grindy a reviewer is, if they post a screenshot of an email written by the sole proprietor of the business telling them to fuck off and die or something similarly toxic, don't you think a lot of people would consciously avoid doing business with a person like that? It probably wouldn't help.
That is what happened where I use to work. They got rid of the dead beat clients, the ones who were always later in payments, the ones who always asking the salesmen for no costs extras to make the sales.
"fix it!" -- "please give more details" or "sorry, this feature is not a near term deliverable" (which is a polite request to get lost). "I am going to a competitor" -- "we think our product is better, but encourage users to try alternatives", etc. My 2c.