And these will go away into eternal [aionios] punishment, but the righteous into eternal [aionios] life.
First, the Greek word [aionios] is used numerous times in the New Testament. In addition to being used for an eternal covenant (Heb. 13:20), an eternal kingdom (2 Pet. 1:11), an eternal dominion (1 Tim. 6:16), an eternal gospel (Rev. 14:6), an eternal redemption (Heb. 9:12), an eternal salvation (Heb. 5:9), an eternal inheritance (Heb. 9:15), an eternal comfort (2 Thess. 2:16), an eternal weight of glory (2 Cor. 4:17-18; cf. 2 Tim. 2:10; 1 Pet. 5:10), and an eternal house (2 Cor. 5:1), it is also used three times to describe God's eternal nature (Rom. 16:26; 1 Tim. 6:16; Heb. 9:14), numerous times to describe the unending happiness of the redeemed (Rom. 5:21; 6:23; 1 John 1:2, et. al.), and five times to refer to the punishment of the wicked (Matt. 18:8; 25:41, 46; 2 Thess. 1:9; Jude 1:7).
In Matthew 25:46, the identical word "eternal" (Greek, aionios) is mentioned twice: once in regard to the righteous, and again in regards to the unrighteous. So, if the punishment mentioned in Matthew 25:46 is merely temporary, as Annilhilationists believe, and if they are consistent in their interpretation method, then heaven, God's covenant, God's kingdom, God's dominion, the gospel, the saint’s redemption, the saint's salvation, the saint's inheritance, the saint's comfort, God's and the saint's glory, the saint's eternal home, the saint's happiness, AND God's very nature are all temporary too! If Annihilationist don’t believe in the necessary conclusion of their interpretation, they need to abandon the false doctrine of Annihilationism.
In Matthew 25:46 the contrast is clear that there is a different quality of existence that is endless in both cases. Moreover, when this verse is read in its fuller context, including Matthew 25:41 which speaks of the unbeliever in eternal fire, a conscious state of torment and painful judgment is clearly understood. The "eternal fire" that unbelievers are cast into is the same "eternal fire" the devil and his angels (i.e., "sent ones"; see “Wasn't hell made for the devil and his angels and not real people?” below) are cast into, and where they are tormented "forever and ever" (Rev 20:10). Therefore, in context, eternal must mean forever and ever, endless, everlasting, or all eternity (cf. 2 Thess 1:9). Hell is forever; it is an endless place of torment.
Second, God is eternal. He lives "for ever [aion] and ever [aion]" (Rev. 1:18; 10:6; 15:7). And glory, which includes God's holy and just wrath against the wicked (Isa. 5:16; Rom. 9:22-23; cf. Rev. 18:20; 19:2-3), is to be given to him "for ever [aion] and ever [aion]" (Rev. 1:6; 4:9-10; 5:13; 7:12). Since the righteous in Christ shall reign with him "for ever [aion] and ever [aion]" (Rev. 22:5), then the wicked surely will be tormented day and night "for ever [aion] and ever [aion]" (Rev. 20:10; cf. Rev. 14:11).
Third, "eternal life" means more than a mere eternal existence. A rock has an existence, but it's not alive. Eternal life for the righteous has a living beginning point. So does eternal damnation for the reprobate. John 3:36 says, "Whoever believes in the Son has eternal life; whoever does not obey the Son shall not see life, but the wrath of God remains on him." Here, as in other places, eternal life is spoken of as a present possession (John 5:24; 6:54; 1 John 5:11, 13). As eternal life is a present (John 17:3) and future reality (Matt 25:46), so the unbeliever is under the living wrath of God now (John 3:36) and in the future (Matt. 25:46). And eternity has already begun and will never end. The wrath of God upon the unbeliever has already begun; they are "condemned already" (John 3:18).