What did Kennedy, Johnson, and Nixon have in common, and what set them apart from each other?

What all three presidents have most in common is the Vietnam War. Due to his idealization in public memory, many people often forget that it was John F. Kennedy who initiated the United States' role in Vietnam. Kennedy believed in the domino theory, which would later be touted by President Richard Nixon's secretary of state, Henry Kissinger—that is, the notion that the success of Communism in one part of the world would lead to other nations falling to Communism. The only solution to what presented a problem for the capitalist West was to eliminate Communism wherever it took root in order to dissuade other nations from adopting Communist rule. This interest in disrupting Communism explains Kennedy's disastrous Bay of Pigs invasion, which was an attempt to overthrow Fidel Castro in Cuba.
Kennedy may have been sympathetic with the civil rights effort (James Baldwin, for instance, was a guest at the Kennedy White House). However, he was slow to act. He regarded himself mainly as a foreign policy president and did not make any move to deal with civil rights until 1963, close to the time of his assassination.
President Lyndon Johnson is actually responsible for the passage of the most important civil rights legislation, including the Civil Rights Act of 1964 and the Voting Rights Act of 1965, overriding the Southern Coalition—a group of powerful incumbent Southern senators—to pass both laws. While signing the legislation, Johnson famously noted that the Democratic Party would lose the South forever. He was right; the South became solidly Republican as a result. Kennedy, while running for president, was still able to court the Southern states and the Dixiecrats despite their skepticism toward his Catholic faith.
Under President Johnson, the United States officially entered the Vietnam War in 1964 after the Gulf of Tonkin incident. Supposedly, the North Vietnamese fired on a United States naval ship, prompting aggressions. This has never been confirmed, however. Johnson was determined to win the war, which was becoming increasingly unpopular with his party. It is for this reason that Johnson did not seek the presidential nomination in 1968, which ended up going to Hubert Humphrey.
Humphrey lost the 1968 election to Richard Nixon. The latter's authoritarian image appealed to "the silent majority": white, middle-aged voters, particularly in suburban areas, who worried about how violent radicalism and the unrest in major cities would impact their lives. Nixon escalated the Vietnam War, rather senselessly, by carpet-bombing Cambodia. Some historians believe that this act prompted hostility against the West, thereby making it easier for the Khmer Rouge to come to power.
Under its leader, Pol Pot, the Khmer Rouge unleashed violence that resulted in the genocide of an entire generation. The massacres of "the educated class" did not end until 1977, after an invasion by the Vietnamese army put an end to the Khmer Rouge's rule.
One could argue that all three presidents, despite their differences in ideologies and political affiliations, were very ambitious and sought to institute regime changes in parts of the world that they barely understood.
https://www.britannica.com/event/Vietnam-War/The-U-S-role-grows

Kennedy, Johnson, and Nixon were all presidents during the Vietnam era in American history.  All three presidents had strong civil rights records and helped end segregation.  
Kennedy and Johnson were both Democrats whereas Nixon was a Republican.  Johnson was a long-time senator before he was picked as Kennedy's vice-president.  Nixon was vice-president under Eisenhower and lost the 1960 election to Kennedy before he won the White House in 1968.  Kennedy won the 1960 election before he was assassinated in 1963.  Kennedy also had the shortest political record of the three, having served just one term in the Senate.  Nixon resigned in 1973 during the Watergate scandal.  Kennedy is best known for his role during Bay of Pigs fiasco and the Cuban Missile Crisis in which he opened up better relations with the Soviet Union.  Johnson is best known for his role in escalating the Vietnam War.  Internationally, Nixon is best known for bringing about a closer American relationship with China.