What does Christianity say about the eternal state of our bodies?
One of the beliefs about the body that was prevalent in Paul's time came from Plato. It viewed the body as a necessary and temporary evil that imprisons the human soul; and the sooner it could be shed then the sooner a person could reach a more perfect state.
That is profoundly different than the way that God designed us and the Bible describes us. The body is not a mere prison of a soul which wants to get free. The body is the means by which the spirit, the soul, and the heart express themselves through visible activity in the material world. And God wants to be made glorious in our bodies. Paul longs in Philippians 1:19-20, "O that I might magnify Christ in my body, whether by life or by death." So the body is a God-given instrument for magnifying Christ.
If we lose the body forever we lose a God-ordained instrument for making much of God. Therefore the future of the body, as described in the New Testament, is that the body will be redeemed. It dies, goes into the grave, it decomposes; but when Christ returns he raises the body. He gives us a body like his resurrected body. He reunites our personality to the body. He makes it whole and complete, and he wipes away every disease, depression and discouragement. And we enter a new heaven and new earth with this body. Therefore it becomes an instrument of praise forever.
You can't say with the Platonists that the body is a prison or a necessary evil. It's not. It's an instrument of worship.
If God is going to heal us in the end, why doesn't he just do it now?
That seems to be part of the larger question of why there is a history of redemption at all. Why didn't God just kill off Adam and Eve after they fell and start over again with a redeemed humanity with no millennia-long struggle with sin and corruption? Why didn't he do it that way? He could have!
He is going to snap his fingers one day and extinguish all evil on this earth. He'll snap his finger and make us holy that day too. And if he can do it someday he can do it now. So why this long, drawn-out battle with sin and disease?
I believe that there are things that God wants to reveal—about the nature, depth, and depravity of sin, and about his own patience, love, and wisdom—that are better displayed by a long history of redemption and a lifetime of struggling out of sin and towards holiness. These things are better displayed through time than if he had simply, in an instant, taken all of my sin away and made me holy the day I got converted, or taken all of redemptive history away and made mankind holy one hour after Adam and Eve fell.
God plans that there be a long history of redemption and a long battle in every individual's life, because in that battle certain elements of his glory, patience, wisdom, love, kindness, wrath, and justice are displayed in ways they would not be displayed if he redeemed things instantaneously.