No. My children receive Holy Communion at our Eastern Catholic church, but they were chrismated in the Orthodox church–just as they would have been in the Eastern Catholic church. At least that is how our priest explained it. Eastern Christians typically receive baptism, confirmation/chrismation and communion all at the same time, as infants or new converts.
This is not so in the Latin Rite. If you are attending an Eastern Catholic church, be advised that unless you indicate otherwise to the priest, he will offer the Eucharistic spoon to your child-- that at least is what our priest does. So you will want to speak to him, if you can, before the Liturgy, or somehow indicate your child has not been confirmed or is Latin Rite.