Regarding the biblical phrase 'everlasting life', would Catholics, Protestants and Orthodox be able to explain how everlasting life destroys death? In the New Living Translation Study Bible notes for Daniel chapter 12, the topic 'Resurrection' on page 1421 offered this explanation of the phrase, 'everlasting life' in verse 2:


  "God's people will finally be rescued from death, not just from human
  kings, fire, or lions. Everlasting life destroys death."


Please note, I am NOT asking for views as to where everlasting life is believed to be enjoyed, or about views on the doctrine of the resurrection. I simply want biblical exegesis regarding the claim that 'Everlasting Life destroys death' as stated in the NLT Study Bible notes.
Yes, that is a theologically sound phrase, and here's why.  From a Christian perspective, it is worth noting that the only time the expression “everlasting life” appears in the Old Testament is in Daniel 12:2.  The relevant verses are Daniel 12:1-2 and 13, a prophetic vision regarding The Time of the End:


  “At that time Michael, the archangel who stands guard over your nation [Israel], will arise. Then there will be a time of anguish greater than any since nations first came into existence. But at that time every one of your people whose name is written in the book will be rescued.  Many of those whose bodies lie dead and buried will rise up, some to everlasting life and some to shame and everlasting disgrace...  As for you [Daniel], go your way until the end. You will rest, and then at the end of the days, you will rise again to receive the inheritance set aside for you.”


The prophet Isaiah spoke of a future time when the shadow of death that hangs over the earth will be removed, when God will swallow up death forever (Isaiah 25:6-8).  The resurrection promise was known to God’s people before Jesus’ own death and resurrection.  After Jesus’ friend Lazarus died, his sister Martha acknowledged that he would rise at the last day.  Jesus then made this bold declaration:


  “I am the resurrection and the life.  Anyone who believes in me will live, even after dying.” (John 11:24-25).  Outside the tomb, Jesus shouted: “Lazarus, come out!” (John 11:43).


It is one thing to claim to be the resurrection and the life, but Jesus proved it by raising Lazarus, who had been dead for four days.  Death has no dominion over Jesus who is life itself, nor does death have dominion over those who are in him (1 Corinthians 15:54-55). Because he has conquered death believers are assured that death shall be destroyed, and in its place will be eternal life.

1 Corinthians 15:26-27: “For Christ must reign until he humbles all his enemies beneath his feet.  And the last enemy to be destroyed is death." 

1 Corinthians 15:54-55: Then, when our dying bodies have been transformed into bodies that will never die, this Scripture will be fulfilled: “Death is swallowed up in victory.  O death, where is your victory?  O death, where is your sting?”  (This quotation is from Isaiah 25:8.)

John 10:15-18 shows how death did not come to Jesus.  He came to it, to destroy it.  Death had no claim on him as he was sinless, so Jesus gave up his spirit at the right time, after he had fulfilled all the scriptures (John 19:24). The only thing left to do to complete the work given to him was to die. John 19:30 shows how Jesus chose his moment of death thereby defeating death: 


  “When Jesus had tasted it, he said, “It is finished.” Then he bowed his head and released his spirit.” 


Revelation sheds more light on the two groups in Daniel’s prophetic vision who are raised to life.  Those whose names are written in the Book of Life are blessed.  “Blessed and holy are those who share in the first resurrection. For them the second death holds no power...”  (Revelation 20:6)

At the final judgment, books are opened (see Psalm 56:8 and Daniel 7:10) and after the dead are judged, death and the grave are thrown into the lake of fire which is the second death (Revelation 20:14).  This is the moment when all of God’s enemies are defeated and destroyed, including the final enemy – death.  Jesus, who is the resurrection and the life, has defeated death and will finally destroy death.  He said “I am the way, the truth, and the life” (John 14:6).

Because Jesus Christ IS The Life and he triumphed over Death with his resurrection, he enables life everlasting.  Everlasting life, the gift of God to all who come to saving faith in Christ Jesus, destroys death.