Translation:My greatest wish is for her to be happy.
"My greatest wish is that she be happy" is standard English (not "Carribean"). The "be" is present subjunctive, as Vladimir points out here.
However, especially nowadays, native speakers who are less familar with standard English have a tendency to reject the subjunctive and to "soften" it by inserting "would" in front of the "be". That is, they reject the subjunctive in favor of the conditional.
These speakers may also reject the classic "If she had come" in favor of "If she would have come".
Some speakers find both these superfluous insertions of "would" quite ugly.