3 Nephi 20:8

John chapter 4 is the chapter were Jesus tells the woman at the well
Whosoever drinks of this water shall thirst again:
But whoever drinks the water I give him will never thirst. Indeed, the water I give him will become in him a fount of water springing up to eternal life.

John, chapter 6 is the chapter where Jesus tells his followers
He that eats my flesh, and drinks my blood, dwells in me, and I in him.
As the living Father hath sent me, and I live by the Father: so he that eateth me, even he shall live by me.
This is that bread which came down from heaven: not as your fathers did eat manna, and are dead: he that eats of this bread shall live forever.

And those statements in chapter 6 sounded so unusual that many people thought he was just too weird, and 
From that time on, many of His disciples turned back and no longer accompanied Him.…

So what is this never being thirsty or this receiving enough sustenance to be able to live by eating and drinking him?

3 Nephi 20:8

And he said unto them: He that eateth of this bread [he had just administered bread and wine to them in sacramental form] eateth of my body to his soul; and he that drinketh of this wine drinketh of my blood to his soul; and his soul shall never hunger nor thirst, but shall be filled.

I think that the message is this:

That it is the spiritual hunger of the soul that He fills and satisfies.

That the living of the laws he taught may make your life better and your sense of accomplishment increase.  They may cause you to do more good in the world and be more dedicated to good causes.  They may help keep you out of trouble and make you more energized and help you feel more enlightened and on track.  They may keep you busily occupied all your life long.

But that living those laws. though extremely worthy and important and good, will not eliminate spiritual hunger.  Only when we make Him part of ourselves, taking into our very souls who and what he is and what he taught, do our souls cease their sense of insufficency and, instead, begin to feel brim-full.

Or, as he said later, at the passover supper:

I in them, and thou in me, that they may be made perfect (complete and whole) in one.