Someone asked me recently, why do you have to be in union with the Church and not in mortal sin, in order to receive the Holy Eucharist?
For the same reason you have to be married, and not have some grave issue unresolved between you and your spouse, in order to have marital relations. Or do you think it's OK to have sexual relations without a true commitment?
Jesus is our spouse: if you aren't committed to him in his Body, the Church, why do you expect to have the intimacy of sharing his Body and Blood in the Eucharist? Imagine approaching a person of the opposite sex and saying, I want to have marital relations with you -- but I do not want to be married to you; because while I like this and that about you, these other things about you? I reject those! Do you think that would work? Do you think it should be acceptable to do that?
So why is it acceptable to approach Jesus and say, yes, I want to have the intimate union of the Eucharist with You, but I do not want to be committed to you! And while I like this and that about You, Jesus, I don't like these things you teach and ask me to observe. I prefer a pick-and-choose approach, how is that, Jesus?
So it's like this: in order to take part in the Holy Eucharist, one must be baptized and believe what the Catholic Church believes. In some rare cases, this can include non-Catholics, but generally not. It includes Orthodox, because in the judgment of the bishops, the issues separating Orthodox and Catholic are minor enough. But many other Christians -- who love Jesus sincerely -- nevertheless have profound differences with the Catholic Faith; including not knowing or believing that the Mass truly is a Sacrifice, that God's grace works through the sacraments, and that the Eucharist really is Jesus' Body and Blood.
But anyone who wants to is, indeed, welcome to receive the Eucharist! But first things first: believe and be baptized -- or if baptized -- be reconciled with the Church, which is Jesus' Mystical Body. Don't try to be with the spouse before you are married!