"Eucharist" was used exclusively for the Presence of Christ in the Body and Blood in the Early Church. "Holy Communion" is a much later term.
But, it doesn't matter. Both are acceptable, and to judge one's orthodoxy on the usage of terminology is, well, silly.
Canonically, they are NOT the same. The correct term, for instance, is "extraordinary minister of Holy Communion" not "Eucharist." Only a priest or bishop can confect the Eucharist. When the laity receive it, it is Holy Communion.