The two events did not share causes, only timeframes. (October-November 1956).
The Invasion of Hungary was led by the Soviet Union in an attempt to quell the Hungarian Uprising of 1956. At the time Hungary was a member of the Soviet-led eastern bloc, but many Hungarians aspired to freedom and democracy and led a spontaneous revolt to take back their country. The Soviet Union would not abide this mentality and promptly invaded Hungary and killed between 2,500-3,000 Hungarian civilians before "restoring order" by putting the Communist autocracy back in power.
The Arab-Israeli War of 1956 (Commonly called the Suez Crisis) began when British and French forces, which were jointly controlling the Suez Canal were forced out by Gamal Abdel Nasser during his attempt to nationalize the Canal. The British and French were not interested in losing the canal to the Egyptians and neither were the Israelis. The British and French motivation was to avoid having to pay tolls and worry about crossing easily. The Israeli motivation was to be allowed to continue crossing between the Mediterranean and Red Seas as Egypt did not recognize Israel at the time. Britain and France incentivized Israel (and joined with Israel) to attack Egypt and take the Suez Canal back. Israel successfully conquered all of the Sinai Peninsula and even landed a foothold on the west bank of the Suez Canal (which meant that they effectively controlled it). The United States, however, intervened and requested that Israel withdraw to the 1949 boundaries under the condition that Egypt return the Canal to British and French joint control.
However, it is worth noting that many scholars in Eastern European studies have argued that had the Suez Crisis not taken place and diverted US attention away from Eastern Europe, it is possible that the US would have stood up to the Soviet Union and protected Hungary's right to freedom. If this had happened, it would have set a precedent for the possible liberation of the Eastern bloc countries four decades in advance of their actual liberation.