I could type a whole bunch but it has already been done for me. There is a good explanation here:
http://byzcath.org/Faith-and-Worship/Who-Are-Byzantine-Catholics.htm
And, you all are NOT given the Eucharist in an Eastern Orthodox Church, anymore than a Baptist or a Methodist is allowed to do so.
At one time in the past, your group may have been Eastern Orthodox Christians but somehow, you all wound up in the Roman controlled sphere of the Pope in Rome. If that is what your folks wanted, well, that is AOK for you... but you all are really VERY different theologically from the Eastern Church.