Yes most definitely!
I have a wild Eugenia on my property which right now has some nice red berries on it. It could be white stopper, not sure.
Then there is Guavaberry (Myrciaria floribunda) although technically not a Eugenia. It is highly regarded here and makes a wonderful rum. A 5 gallon bucket of this fruit will fetch $500 easily.
We also have a type of Eugenia here called "bunch berry." At least that's what they call it in the Bush Skills class my son takes at:
www.mtvictorycamp.com This Eugenia grows in the understrory of our wetter parts of the island. The wood is very hard and beautiful. In the Bush Skills class they cut the trees down (leaving the stumps so they can grow back), strip the bark and use the poles for building shelters. Burn the part that goes into the ground fiurst and they last a long time. The mother in law tongue grass (Sansevieria trifasciata) and a type of mallow are beaten to make cordage and used as lashings for the poles. Then Royal Palm husks are used for the roof. Makes a sturdy, element-proof shelter that will last for awhile.
If anyone has an interest (or has kids with an interest) in learning primitive skills in an awesome tropical environment, I highly recommend the folks at the Mt. Victory camp. Awesome people who studied under the master Tom Brown.
Some other things unique to the Caribbean: Gooseberry wine, sea grape wine, Mama Juana (real big in DR), Mauby drink made from the bark of the Mauby tree.
We have a festival every few months down town. This lady comes in her beat up pickup loaded with coconut and sugar cane. She presses the sugar cane right there and then mixes the cane juice with fresh coconut water. The combination is so good! If you want to reinforce it, she has her arsenal of a bar too so you can get all kind of different rum or other libations added in- just be careful she's a bit heavy handed on those libations.