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What is the Eastern rite Catholic version to the Western or Latin rite chant of the Exuitent which is done on Holy Saturday night/vigil to Easter Sunday?

The celebration is quite different. The following was posted by someone who visited the Maronite Church in Washington DC.

There were some unusual aspects of this Easter Vigil. There was incense but no use of Holy Water or candles. The Maronite Church tends to baptize their Catachumens on the Feast of the Epiphany in January. Candles are not as import of an symbol at the Easter Vigil, as the Maronite Church breaks fast on noon of Great Saturday during the simple "Awaited Light" ceremony. This may explain why there is not pent up anticipation for the Easter Vigil as observed amongst the Maronites.
It was remarkable how much this Liturgy celebrating the Resurrection emphasized the Glorious Cross. The Chorbishop made prayers on the four corners of the altar with the processional cross, as if to proclaim the hope of the resurrection to all the Earth. The faithful were invited to venerate the Glorified Cross as they received Communion.
Another interesting Easter feature of this vigil Qurbono was the emphasis on the Empty Tomb. As the faithful departed from the Divine Liturgy, they were given flowers from the Empty Tomb as well as an Easter Egg.

Her description is fairly representative of that liturgy which takes place at Midnight in all Maronite Churches. Note that on Good Friday, parishioners bring flowers to the "funeral" for that is what we celebrate - the death and burial of Jesus. Those flowers are then redistributed on Easter, first fruits from the Risen Lord.