The two countries with square flags are Switzerland and the Vatican. However, the Swiss flag can be seen at the Olympic games with a 2:3 rectangular proportion that matches the other country's flags.
The Swiss flag is square and red in color with a white cross. At sea, however, the Swiss Naval Ensign is used; it has 2:3 rectangular proportions instead of the 1:1 square proportions. The flag is kept in standard square proportions when it is flown outside of the U.N. headquarters in both New York and Geneva.
The flag for the Vatican is divided vertically with yellow on the left side and white on the right side. Within the white right side, there is an emblem that has two crossed keys with a papal tiara. The emblem is a reference to the keys that are mentioned in the New Testament of the Bible and symbolizes either: St. Peter's access to the kingdom of heaven or the dominion of the papacy over temporal and spiritual matters. The Vatican flag used to be red and gold but was switched to the yellow and white stripes during the early 19th century. It also used to have representations of Saint Peter and Saint Paul.