I can refute it for you. HAL does not have a topless area to sunbath on it's ships and they are not very open to the idea. You may see topless if you look very hard on the back side of Cozumel at the remote beaches or if you take an excursion to Playa Del Carmen. You definately will not see any toplessness in GC as anyone doing so there would be arrested, they are very Victorian in attitude there. In Montego Bay you will most likely see some topless at the public beaches but total nudity is pretty much restricted to the adult resorts.
Now, the big thing for your friend to understand that these folks that choose to doff their tops or even those that opt for au natural are not perverts or loose morals or probably even concerned with anything sexual. Differant cultures have differing mores and one needs to not just be aware of them but really understand that not everyone thinks the way we Americans have been brought up to think.
I would be surprised to find topless behavior on any of the stops you mentioned, especially Cayman. Public modesty is the best policy on that island. If you are at the beach in your swimsuit, that is fine, but you should have a wrap at least when you are going or coming.
i would doubt that you would see any nudity at any of the ports you mentioned. Jamacia maybe at the inclusive resorts but that's about it. As for the HAL cruise ship, most definatley not. I don't know where you or your friend got this information but it is incorrect. For future reference its the french islands that mostly go topless, total nudity is not prevalent, its no big deal, just different cultures
You must realize that the public nudity you might see is not a Playboy photo shoot, but just a mix of average people some big, some small, etc. I remember being at Club Med in Morroco back in the 1970's as a child and seeing the topless sunbathing. It really is no big deal and if you do see it, just mark it off of one of your life experiences...it may give you a laugh or two later on down the road.