I'm not going to mark this as the correct answer, because I'm not sure there is a correct answer. However, I want to attempt to answer it myself, just to add to the discussion.
Imagine a conversation where Mr. A is complaining about the Demopublicans, media corruption, the vanishing middle class and on and on. Mr. B replies "If you don't like it, why don't you move to Cuba?"
Sometimes, the response is "somewhere else," but the Mr. B's often name a country that is widely loathed by right wingers and/or is widely regarded as a terrible place to live for one reason or another. For example, Cuba is generally poor and authoritarian.
Whether or not this qualifies as a fallacy depends on both the context and the precise wording, particularly the words "why don't you move," which can be loosely translated "you SHOULD move."
Mr. A: "Thanks to capitalism, I'll never retire!"
Mr. B: "If you don't like capitalism, you should move to Cuba!"
There is an element of cherry picking in B's response. Instead of Cuba, he could have suggested moving to a socialist state that isn't being targeted by the U.S., like Sweden.
One might also argue that there's an element of - I'm not sure what to call it - obfuscation or over-complication? In fact, there are many reasons why a U.S. citizens might not want to move to Cuba or China, including the cost of moving and the fact that most U.S. citizens don't speak Spanish or Chinese. It almost sounds like a straw man argument, particularly because the discussion wasn't originally about Cuba or China.
Or you could accuse Mr. B of dodging the question. Instead of engaging in an intelligent discussion about capitalism, democracy, or whatever, he suggest that the argument can be boiled down to a simple comparison between the U.S. and the Cuba.
It has already been noted that such a question could be construed as a false dilemma fallacy or even a simple insult - an ad hominem attack.
Finally, the "why don't you move" retort could be an example of all the above, a classic example of killing two birds with one stone.