Diplomatic pressure, economic sanctions (although in 1956 not so sure how they would work, by 1967 Soviets were buying food from the west), sure. Actual military intervention on behalf of the Hungarians or Czechs, nope. Had there been a communist insurrection in Greece, for example, in 1956 and the Soviets had entered Greece across the Bulgarian border, what do you think the NATO response would have been? The pro-Soviet elements in Hungary (and Czechoslovakia) were seen as the "legitimate" government of those countries requesting "fraternal assistance" from counter revolutionaries and NATO/US military intervention would be seen as an invasion. For the WP as well NATO the mantra was "an attack on one is an attack on all".
Had NATO/US forces entered Hungary or Czechoslovakia very rapidly you'd have US and Soviet forces shooting at each other. That will almost certainly escalate. In 1956 this would end badly for the USSR although the USA and especially Europe would take a hit. In 1967, things suck for everyone although worse for WP than NATO but still...
With the Suez situation, the UK & France busy with Egypt and the USA at odds with Britain and France, even had Ike wanted to pressure the Soviets in 1956 over Hungary he was not in a good position to do so. In 1956, putting a lot of pressure on the USSR short of war is only possible if Suez does not happen. In 1967 the USA is completely bogged down in Vietnam and is a really bad position vis a vis the USSR for any potential conventional military confrontation meaning nukes would tend to fly early. As long as the USSR was willing to act the way they did to suppress anti-Soviet elements in Hungary and Czechoslovakia neither the USA nor NATO is going to riskj a general war, especially with nukes, to support them.