And only God will judge who is "born again!"
No, non-believers will not have "eternal life," but will have eternal "death."
What is the "eternal life" of which Christ spoke?
It is an awakening as a spiritual being -- no longer dependent upon the physical for perception and action. This means that you no longer need your Homo sapiens body for the ability to see and the ability to act upon the physical universe or to control your location within or without the physical universe.
Right now, most people have limited continuity of consciousness. This is caused by the fact that spirit alone can no longer see. This is the "darkness" of which the Bible speaks. The spirit depends upon the body's eyes and the body's brain for consciousness. Without them, the spirit is wholly asleep ("dead" to the world). This happened to each one of us individually in the Garden when we ate of the fruit (product) of the tree of knowledge of physical dichotomy -- good-evil, right-wrong, generous-selfish, wisdom-stupidity, compassion-indifference, confidence-doubt, and many more. These are what blind us -- the action-reaction dichotomies of ego.
When someone is born again, they are no longer a physical body with a dead asleep soul; they are a child of God with a Homo sapiens body. When the body dies, the consciousness continues unabated, because the true self no longer depends upon the "graven image" of human life -- but depends instead on the source of all life.
When Jesus said that those who try to save their lives will surely lose them, he was talking about ego. Ego is a physical cog -- false self -- and the self that must die in order to gain the awakening of spirit and everlasting life.
Humility is the antidote to ego. Christ made this point clear when he said that the meek will inherit the Earth, and he who is first will be last, and he who is last shall be made first.
Non-believers will remain trapped in the physical realm and wholly dependent upon it. When the Heavenly Host withdraws from the rescue mission, as declared in Genesis 6:3, those left behind will not have the assistance with which we have been blessed. And when Earth begins to bake from the aging sun, all will become as a lake of fire, the bodies will be burned alive and Earth with it.