The creation account in the beginning of Genesis is really an elucidation of God’s determination to make mankind in his image. The creation of the universe itself and all other lifeforms is treated as the backdrop to that ultimate aim. No more explanation than “and God said…and God saw that it was good” is offered for all of those creations, but for mankind a bit more needed to be said. The thrust is that mankind is unique, special among God's creations with something nothing else in all the physical world has.
Mankind was made in the image of God, which means they are a likeness resembling God. God is non-corporeal and outside of the created order (John 4:24; 1 Timothy 1:17), which means that man’s resemblance to God is not physical but something else. Physically, mankind is much like anything else that is alive and is separated by mere degree from all else. However, nothing else is like mankind in those areas of divergence noted above and that is where the image of God shines forth. God is the only thing other than man (and angels) that shares those qualities.
Presumptively, God fashioned man, physically, from the same material he had used to make other creatures. Whereas they were brought forth from the earth by a mere word, mankind was formed [Hebrew: yatsar] by God in the manner of a potter and then directly breathed into by God which granted man soulish life. Although later in the creation account, that word (formed) was applied generally to all the creatures God had made, I find it interesting that in dealing with the detail of creation, a clear difference in how that played out is specified.
Whereas a general, creative word was sufficient for every other creature, with man God got his hands dirty and infused his own breath into Adam. Every creature had living being [Hebrew: nephesh chayyah
What is important about this distinction, it seems to me, is that the quality that makes mankind’s living souls uniquely from God is also the means by which God’s image was uniquely communicated to man. We are in God's image, not just because we are like God descriptively, but because we came directly from God substantively. We are, in essence, breath from God. Human beings truly are the offspring of God.
As wonderful as that is, it has a drawback--it means we last forever, just like God. God is eternal and the breath that came out of God and was put into man (and made him a living soul) lasts forever too. Therefore, people never cease to exist, their soul is eternal. Ultimately, body and soul will brought together, as at first, and permanently assigned to their place of eternal abiding. So the only question about our future existence is not if we will, but where we will and under what conditions.
We certainly can't be destroyed, anymore than God can!
Most of us are only all too aware of our need for the redemption our broken, dying bodies: physical death, and what leads up to it, is enough to get that message across. Thankfully, we have the necessary vicarious sacrifice in the death of Christ and his victorious resurrection from the dead for that, provided we place our trust in him. But what is the more essential need included in the mix is the redemption of our eternal souls, those whisps of the very breath of God which last forever. Everlasting life is in our hands from the scarred hands of Christ, if we would just pause in faith and catch our breath.