No, the US did not invade Cuba and does not claim (other than GITMO) any part of Cuba as ours. Oh, and before you go crying about the Bay of Pigs - that was just a bunch of disgruntled Cubans that wanted their country back.
Now, IMHO, we should have invaded Cuba, taken Cuba, eradicated every last Cuban on that island, and hung Fidel from the Gallows. But, I wasn't even alive then, and I doubt that philandering Kennedy would have listened to me anyway.