Do any countries officially use the Julian Calendar for fixing the date of Easter?
The Ukraine (at the moment of answering this question) uses the Gregorian Calendar for its' international affairs. However, it officially celebrates Easter according to the Julian Calendar. How long this will last is anyones guess. The same is true for its' celebration of Christmas.
Ukrainian MPs drafted a law that proposes to celebrate Christmas in Ukraine as a national holiday both 25 December (Gregorian calendar) and 7 January (Julian calendar). They believe that their initiative will not only take into account the religious need of many Protestant and Roman Catholic Christians, but also will unite the Ukrainian society.
The draft proposes to amend Article 73 of the Labor Code of Ukraine, adding 25 December (Christmas) to the list of holidays. Seventh of January is already in the list. This means that Ukrainian will celebrate Christmas and 25 December and 7 January.
MPs argue that according to statistics from the Ministry of Culture, in addition to about 23,172 Orthodox, Greek Catholic and some other Christian communities that celebrate Christmas on January 7 (according to the Julian calendar), today in Ukraine there are about 11,000 Catholic and Protestant communities (about 30 % of all religious organizations in Ukraine) that celebrate Christmas on December 25 (according to the Gregorian calendar). - Christmas According to Gregorian Calendar may become a national Feast in Ukraine
Traditionally, the Byzantine Catholic and the Eastern Orthodox churches use the Julian Calendar to calculate their feast days, including Easter. The Orthodox Easter is a legal holiday in the Ukraine.
Counties that that celebrate the Orthodox Easter as a holiday are as follows:
The Coptic, Ethiopian and Eritrea Orthodox Churches use the Alexandrian Calendar for their liturgy.
The Coptic calendar, also called the Alexandrian calendar, is a liturgical calendar used by the Coptic Orthodox Church and still used in Egypt. This calendar is based on the ancient Egyptian calendar. To avoid the calendar creep of the latter, a reform of the ancient Egyptian calendar was introduced at the time of Ptolemy III (Decree of Canopus, in 238 BC) which consisted of the intercalation of a sixth epagomenal day every fourth year. However, this reform was opposed by the Egyptian priests, and the idea was not adopted until 25 BC, when the Roman Emperor Augustus formally reformed the calendar of Egypt, keeping it forever synchronized with the newly introduced Julian calendar. To distinguish it from the Ancient Egyptian calendar, which remained in use by some astronomers until medieval times, this reformed calendar is known as the Coptic calendar. Its years and months coincide with those of the Ethiopian calendar but have different numbers and names. - Coptic calendar (Wikipedia)
The Coptic Liturgical Calendar celebrates Christmas and Easter on the same dates as the rest of the Orthodox Churches and countries using the Julian Calendar for religious reasons.