Not to get too deep into Christology, but perhaps some study of the hypostatic state should be presented. I have a human spirit, and before I was regenerated, that human spirit was "dead" in that it was separated from God because of the consequence of Adam's sin. So when I was born again, my spirit was placed "in Christ" and so together with Christ I was made alive spiritually, no longer separated. That pretty much covers the idea, in my opinion, of humans being born again in Christ.
Think that the Bible clearly points out that all humans have a sin nature inherited from Adam in us from birth, and thast Spiritual death is due to us being in same state Adam found himself in after he sinned and fell before God...
When we are "born again" by God, Holy spirit resides within us, so we now have 2 natures, old sinful "flesh" from Adam, new nature from Christ...
Back to basic Christology. Was Christ part God and part Man? Nope, He was 100% God and 100% Man.
Did Christ have two spirits, a human spirit created at conception, and a divine spirit? Nope, only a divine Spirit, the second person of the Trinity.
Jesus had/ and STLL has 2 natures residing within His being, one is divine nature of God, other is sinless human nature, as Adam had before he chose to fall...
Was this God Spirit ever separated from the Trinity? Nope, so the Spirit of Christ never died, i.e. with dying being equated with being separated due to unholiness. Jesus was sinless on the cross. He did not become a sinner separated from God.
While on the Cross, becoming sin for our sakes, he experienced "hell" seperation gfrom His heavenly Father... believe THAT is what Jesus know would be worst aspect to the Cross, period of time experiencing seperation from rest of Trinity "My God why have you forsaken me?"
Yes I know some will point to the text that says Christ became sin for us, but according to my Christology, Christ became a sin offering for us, which is a variant of the text.
When Jesus gave up His Spirit (not Spirits) He physically died, He "tasted death."