Details are important here. There are so many variables that people can't really give answers to until they are defined. The Cuban Missile Crisis developed the way it did because of actions JFK took earlier such as the failed Bay of Pigs invasion and the 1962 Jupiter missile deployments. Change those and lots of things change.
Eisenhower's massive retaliation doctrine was based on the idea that should a conventional war erupt in Europe, the USA would escalate rapidly to nuclear war. This was a warning to the Soviets that there would be no limited war in Europe to dissuade them from attempting to do anything. But the Caribbean is not Europe, and the Soviets can't project power like they could in Europe. No one in the Western Hemisphere was under threat of a conquest by the Red Army. That Ike would nuke the Soviet Union because the Cubans shot down a U2 spy plane is ridiculous. When the Soviets shot down a U2 in 1960, Ike didn't order a nuclear attack on the Soviet Union.
Ike could very well have bombed and invaded Cuba in similar circumstances, but we'd also have to see how things developed to get to that point beforehand. Ike was not Curtis LeMay. He actually was capable of very complex analysis. Massive retaliation was a policy in order to achieve certain aims - to protect Europe and keep the peace in the cheapest way possible so that the American economy would not become burdened. He didn't do it because he was incapable of thought. He would have sought to keep any Cuban war localized (which as we know would probably not be possible, but then again we'd need to know why Khruschev choose to put nuclear weapons in Cuba in the first place). He would not have used nukes in Cuba unless America was attacked by nukes (although at that point a strike against the Soviets would probably have been done). A crisis in the Caribbean was far different than a crisis in Europe.
The matter at hand is not necessarily a Cuban Missile Crisis but rather a similar quagmire situation where the superpowers are looking each other in the eye, with neither able to back down.
In terms of Cuba, it is important to note that the invasion of the exiles failed as it did because the concept as a whole was a Caribbean Sealion. The fact that exiles were training in Guatemala for an invasion was already leaked by the press, and Castro therefore rounded up dissidents (potential or real) and left out dummy planes for the air force to target, and prepared for an imminent invasion to arrive. It would have required direct US intervention to save it no matter where it was, which was a failure of that plan. Such an unprovoked attack would have looked very bad to our Western allies, as well as our Latin American allies, and would have looked like a return to the Banana Wars. As I recall the details, the CIA and military believed that the plan would have required US support, though they did not share this with Kennedy. However, they seemed to believe that when the exiles faltered, Kennedy would intervene directly, just as Eisenhower would. Kennedy did not. If Eisenhower similarly sought a covert operation, he would have received similar false promises, though he would likely have intervened. And that would have caused the problems I mentioned.
However, utilizing the example of Cuba, we can reflect on how Eisenhower would have reacted to a similar situation wherever it may have occurred. With an ongoing Cold War, such a situation is increasingly likely to happen somewhere at some point. In terms of the U2 being shot down, to say Eisenhower would have reacted is certainly not ridiculous. The Gary Powers incident was a world away from the shooting down of the U2 during the crisis. In the former incident, it was a covert US action where America denied that such a thing even existed, and it was certainly not a situation where the two superpowers were already in a crisis situation that had the potential of direct conflict. The U2 being shot down was a military attack in that latter situation. It is important to note that the Joint Chiefs that Kennedy inherited were from the Eisenhower administration. Kennedy felt that he did not have the clout to get rid of them and that he would be criticized for doing so, as it would be interpreted as undermining the military wisdom of Eisenhower. His strategy was to slowly ween them off. Following the shooting down of that U2, the military urged Kennedy to launch airstrikes against Cuba. That incident had the prospect of spiraling the situation out of control to nuclear war. One has to understand this looked like the opening salvo in a war. Indeed, throughout the crisis, the military as well as many in the cabinet urged the president to launch air strikes or an invasion. It was Kennedy who took the diplomatic road, and tried to understand the Soviets. This is the military that would be advising Eisenhower. Indeed, Eisenhower urged Kennedy to take similar action. When Kennedy asked if the Soviets would move on Berlin in retaliation, Eisenhower replied they just wouldn't. Indeed, we know now there were already nuclear installations active in Cuba, and that the Soviets would have moved. I will not disagree Eisenhower would have tried to use diplomacy up to a point. However, I do argue he would have reacted with an attack in situations such as the U2 being shot down, which would have spiraled into nuclear conflict, and that he would have pursued an invasion of Cuba to try to "localize" the situation, assuming the Soviets would simply back down, and they certainly would not have.
What made Cuba so dangerous was that Khrushchev read Kennedy as weak, saw an opportunity to promote Soviet interests in the Western Hemisphere, and overreached. In Mao's maxim, he reached out with his bayonet expecting to meet flesh, and instead met steel.
With an Eisenhower or Nixon in charge, it's unlikely that Khrushchev will perceive that degree of weakness, and so there almost certainly won't be an attempt by the Soviet Union to push into areas that the United States considers its' vital security interests. Conflict (not necessarily armed) will be confined to peripheral areas - things like Lebanon, Suez and the Taiwan Straits, which will take the imminent danger out and allow cooler heads to prevail.
He did read Kennedy as weak. However, there was nothing Kennedy could have done or said. At Vienna, Khrushchev was looking for a fight and not discussion. Under Eisenhower, relations were already deteriorated and made as bad as ever by the U2 incident. This set the course of tension in the 1960s for Kennedy to inherit, and would have been a problem for a hypothetical third Eisenhower term. Throughout that period he was taking an aggressive posture, assuming it would make the US back down to Soviet interests. That was a dangerous course of action, which may have guaranteed Western acceptance of the existence of East Germany and the protection of Cuba, but put the world on the brink of war. This is dangerous under Kennedy, just as it would have been under Eisenhower. That is not necessarily a matter of perceived weakness. Khrushchev was behaving recklessly. That lent to his removal from office. With the rise of Castro, Khrushchev was already trading barbs at the United States during the Eisenhower administration. I'll quote briefly a History.com article. However, I reiterate that this is not necessarily about a Cuban Missile Crisis, but some similar situation at some point around the globe. Indeed, if Cuba fell to US/Cuban exile invasion prior to the OTL crisis, not only would it have damaged US reputation in the developing world and among our allies, it could have pushed Khrushchev to seek more overt, aggressive, and obviously reckless action elsewhere in the world, thereby giving us a different quagmire crisis. A damaged US reputation in Latin America and the post-colonial nations would have also strengthened the Soviet position to exploit them to their advantage.
http://www.history.com/this-day-in-history/khrushchev-and-eisenhower-trade-threats-over-cuba
Khrushchev fired the first shots during a speech in Moscow. He warned that the Soviet Union was prepared to use its missiles to protect Cuba from U.S. intervention. “One should not forget,” the Soviet leader declared, “that now the United States is no longer at an unreachable distance from the Soviet Union as it was before.” He charged that the United States was “plotting insidious and criminal steps” against Cuba. In a statement issued to the press, Eisenhower responded to Khrushchev’s speech, warning that the United States would not countenance the “establishment of a regime dominated by international communism in the Western Hemisphere.” The Soviet Premier’s threat of retaliation demonstrated “the clear intention to establish Cuba in a role serving Soviet purposes in this hemisphere.”
The relationship between the United States and Cuba deteriorated rapidly after the Eisenhower-Khrushchev exchange. The Castro regime accelerated its program of expropriating American-owned property. In response, the Eisenhower administration severed diplomatic relations with Cuba in January 1960. A little more than a year later, in April 1961, the CIA-trained force of Cuban refugees launched an assault on Cuba in the ill-fated Bay of Pigs invasion. The invaders were killed or captured, the Castro government cemented its control in Cuba, and the Soviet Union became Cuba’s main source of economic and military assistance.