What non-canonical texts do Eastern Catholic liturgies use?

The Anaphora (Greek-speaking Eastern Christianity - Qudaša in the Eastern Syriac tradition) is the most solemn part of the Divine Liturgy, or the Holy Sacrifice of the Mass, during which the offerings of bread and wine are consecrated as the body and blood of Christ.

Source: https://en.wikipedia.org/wiki/Anaphora_(liturgy)#Armenian_Rite

The Byzantine Rite uses three anaphoras, which are the core part of the Divine Liturgies which take the same name:

• The Anaphora of St. John Chrysostom
• The Anaphora of St. Basil, once used throughout the year, and now used only on some Sundays and feasts
• The Anaphora of St. James

The Armenian Rite, used mainly by the Armenian Apostolic Church, uses currently the Anaphora of St. Athanasius.

The present Coptic Orthodox Church and Coptic Catholic Church have three Liturgies:

• The Liturgy of St. Basil (4th century)
• The Liturgy of St. Mark the Apostle, this liturgy is also known as the Liturgy of St. Cyril
• The Liturgy of St Gregory the Theologian

The anaphoras currently used by the Syro-Antiochene Rite (or West Syriac Rite) are numerous and the main are:

• Anaphora of Twelve Apostles
• Anaphora of St. James, a different anaphora from the Byzantine Rite's one
• Anaphora of St. Mark the Evangelist
• Anaphora of St. Peter
• Anaphora of St. John the Evangelist
• Anaphora of St. Xystus of Rome
• Anaphora of St. Julius of Rome
• Anaphora of St. John Chrysostom, a different anaphora from the Byzantine Rite version
• Anaphora of St. Cyril of Alexandria
• Anaphora of St. Jacob of Serugh
• Anaphora of St. Philoxenus of Mabbug
• Anaphora of St. Severus of Antioch
• Anaphora of Mar Jacob Bar-Salibi

The Antiochene Maronite Church is one of the richest in the number of anaphorae contained in its Liturgy, most of them belong to the tradition of the Antiochene rites. There are at least seventy-two Maronite Anaphorae.

Source: https://en.wikipedia.org/wiki/Divine_Liturgy

For a list of Eastern Orthodox Liturgical books, see: https://en.wikipedia.org/wiki/Eastern_Orthodox_worship#Liturgical_books.

Some books from the referenced page include:

• Menologion: A collection of the lives of the saints and commentaries on the meaning of feasts for each day of the calendar year, also printed as 12 volumes, appointed to be read at the meal in monasteries and, when there is an all-night vigil for a feast day, between vespers and matins.
• Synaxarion (Greek: Συναξάριον; Romanian: Sinaxar): The Synaxarion contains for each day of the year brief lives of the saints and meanings of celebrated feasts, appointed to be read after the Kontakion and Oikos at Matins.
• Patristic writings: Many writings from the Church fathers are prescribed to be read at matins and, during great lent, at the hours; in practice, this is only done in some monasteries and frequently therein the abbot prescribes readings other than those in the written rubrics. therefore it is not customary to enumerate all the volumes required for this.
• Collections (Greek: Ανθολόγιον, Anthologion; Slavonic: Сборникъ, Sbornik): There are numerous smaller anthologies available which were quite common before the invention of printing but still are in common use both because of the enormous volume of a full set of liturgical texts and because the full texts have not yet been translated into several languages currently in use.
• Typicon (Greek: Τυπικόν, Typikon; Slavonic: Тѵпико́нъ, Typikon or уста́въ, ustav): Contains all of the rules for the performance of the Divine Services, giving directions for every possible combination of the materials from the books mentioned above into the Daily Cycle of Services.

For a complete list of non-canonical text references of the Old and New Testaments, see: https://www.aidanorthodox.co.uk/wp-content/uploads/2020/05/non-canonical-texts.pdf.

Of these texts, the Orthodox but extra-canonical text mentioned include:

• Didache
• Epistle of Barnabas
• Protoevengelium of James
• Martyrdom of Polycarp
• Justin Martyr

The PDF link above for the above books, links to detailed information on each of the books listed.

From the links provided, you will be able to gain good knowledge on the Liturgies and books used.