The answer
It is the attribute of infinity, not just eternity (or having an existence after death), which belongs to God alone.
God is infinite, which means that He had no beginning. No other being has this quality, for all other beings have been created. Consequently they had a beginning at some point, and had no existence before that beginning. They are, therefore, necessarily finite, because at some point they did not exist. And since they were created they cannot be infinite.
Eternity in the form of eternal life, however, is something which God has given to some of His creatures.
God created human beings with an immortal soul, both the righteous and the wicked are alike in this respect.
This potential for immortality doesn't mean that human beings are gods, they are still human beings, in spite of the fact that God has bestowed on them eternal life. If eternal life were one of God's attributes alone, it would be impossible for a human being to live on after death and enjoy eternal life, because a human being cannot turn himself into a god.
Existence in an afterlife is for both the righteous and the wicked, but they will differ in their fate, as the Bible tells us concerning the Day of Judgement: "Then they (ie. the wicked) will go away to eternal punishment, but the righteous to eternal life. " (Matt. 26:46)
If we did not believe in this eternal existence for the wicked, we would on one hand be contradicting the Bible, and on the other we would be becoming like the Seventh Day Adventists who believe that the punishment of the wicked is non existence and annihilation.
This painful eternity is also for Satan and his angels.
The Bible says that on the Day of Judgement the Lord, "will say to those on His left, " 'Depart from Me, you cursed, into the everlasting fire prepared for the devil and his angels " (Matt. 25:41)
And the Book of Revelation says about Satan's punishment.. "The devil, who deceived them, was cast into the lake of fire and brimstone where the beast and the false prophet are. And they will be tormented day and night forever and ever. " (Rev. 20:10)
The phrases 'for-ever and ever' and 'eternal fire' mean that Satan and the wicked will live for-ever, but in torment.
Those who belong to the Jehovah's Witnesses and the Seventh Day Adventists, however, deny this.
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Thrice-Blessed Pope Shenouda III of Blessed Memory