Cuban migration to the US has mostly happened over the course of the 20th century, after the Cuban Revolution and the rise of Fidel Castro. So, during the Civil War the Cuban population would have been minuscule.

Post-WWII, with the development of US highways, cars, and cheaper air travel, it seems that people began to travel to Florida for vacation and began to settle there for retirement due to a favorable climate. And, of course, the further south you go the warmer the climate becomes, so it makes sense why a large number of people from other states would end up in the southernmost part of Florida. This has happened to the point that only about 1/3 residents of Florida were originally born in the state.

So that's why you see the cultural mix between the Northeast and Cuba in South Florida.

But since you asked about segregation, it appears that yes, there are instances of holding on to segregation in South Florida (although there was less resistance to the Civil Rights Movement compared to other southern states). Here is a good write up on some of the issues surrounding Miami Beach and segragation.