Question

Why were events in Cuba troubling to the American government?

Answer 1

I will assume you are talking about the cold war post revolution Cuba? - If so then the reason was rampant McCarthyism

Explanation:

In perception of communism in the US in the 50's and early 60's was one bordering on the hysterical - the McCarthy witch hunts are testimony to this - When the pro-US Batista regime in Cuba was overthrown by Castro's Communist forces.
The US tried everything it could be reverse the revolution - a communist country so close to the US was deeply troubling to the US (dont forget the "domino theory" was created around this time) so the US treated cuba as one would someone with a contagious disease....
They funded and facilitated the Bay of Pigs invasion, when Cuba allowed Missiles capable of delivering nuclear weapons to most places in the US - Kennedy had a total sense of humour failure (despite the US having similar missiles deployed on the USSR-Turkish border)n