Though both Paul and John talk about the "body of Christ" in today's readings, each approaches the concept from different directions.
Most of us, as Catholics, are more comfortable with John's theology than Paul's. From childhood we've been taught about his manna symbolism. On the other hand, we rarely hear (or practice) Paul's theology.
The Deuteronomy author sets the stage for John by reminding his community of Yahweh's care for them during the Exodus. "... Yahweh, your God... directed all you’re journeying in the desert... He let you be afflicted with hunger, then fed you with manna, a food unknown to you and your ancestors, in order to show you that not by bread alone does one live, but by every word that comes forth from the mouth of Yahweh."
This feeding image is custom-made for John's account of the institution of the Eucharist. Removing the institution from the Last Supper, he repositions it in the context of Jesus' miraculous feeding of the crowds. (Of course, in its Last Supper place, he inserts another "sacramental sign:" the foot washing.)
No one can doubt John's conviction that Jesus' body and blood is "true food and true drink." Nor can they doubt that our eating and drinking these elements will help us achieve eternal life. "Unless you eat the flesh of the Son of Man and drink his blood, you do not have life in you. Those who eat my flesh and drink my blood have eternal life, and I will raise them up on the last day."
Eternity is the point at which John's manna metaphor breaks down. The author of Deuteronomy knew nothing of an afterlife as we know it. For him, manna was just a help to eventually reach our earthly promised land. Like manna, the Eucharistic bread "... is the bread that comes down from heaven," but there's a big difference. "Unlike our ancestors who ate and still died, whoever eats this bread will live forever." Jesus feeding us now is a dramatic sign that Jesus will feed us for all eternity.
Paul, on the other hand, is much more concerned with the here and now. Though the Apostle still seems to be expecting Jesus' Second Coming to take place during his lifetime, he's worried about some of his communities splitting apart before that earth-changing event. Because unity is by far his constant focus, he envisions the Eucharist as a force which unites us on two levels.
First, "the cup of blessing that we bless, is it not a participation in the blood of Christ?" Second, "the bread that we break, is it not a participation in the body of Christ?"
It's not only essential that we constantly be one with the risen Jesus, but in order to carry out the ministry of Jesus, we also must be one with all those who make up the body of the risen Jesus. Just as, for John, Jesus' body and blood guarantee we'll experience an eternal oneness with Jesus, Paul is concerned that whenever we join with others in celebrating the Lord's Supper, we become one both the person we commemorate and with everyone doing the commemorating right here and now.
One outward sign of this unity is the form of the bread we employ for our meal. "Because the loaf of bread is one, we, though many, are one body, for we all partake of one loaf."
Paul would be appalled to discover we Catholics long ago discarded the practice of using one, big, messy loaf in favor of more manageable individual servings. On the other hand, I presume Paul would be even more appalled at the institutional divisions (clergy/laity) at play during our Eucharists than he would be worried about the type of bread we use. The latter run completely counter to Paul's vision of community.