The original question had to be edited. I think that one specific continuity with the Eisenhower Era that President Kennedy shared was the idea that American strength in the world did not have to be advanced through constant military entanglement. Like Eisenhower, President Kennedy sought to create avenues where world peace could be realized. Continuing from Eisenhower's lead, Kennedy's foreign policy did not have to be rooted in constant antagonism with the Soviet Union. The decrease in the intensity and vitriol in the Cold War that was seen with Eisenhower continued into Kennedy's administration. Even with the Cuban Missile Crisis, there was a desire to "understand" the enemy as opposed to making rash judgment to condemn the enemy. Replacing unending use of the military with an analytical approach that sought to better control the "military industrial complex" was something evident in both administrations. The need to understand geopolitical balance through prudent and cautious use of the military was something that Eisenhower demonstrated and Kennedy continued.
Certainly, the issue of South East Asia proved to be a fundamental challenge to both Presidents. Kennedy continued Eisenhower's installment of advisers in the region. He also continued the previous administration's willingness to lend support in a variety of ways, short of installing military troops in the region. The issue of Vietnam is one in which both Presidents saw a level of importance in its focus. In this, President Kennedy continued what Eisenhower started.