Eternal life is more than endless living. It includes fellowship with God and at least beginning to live life as God lives.
When God gave us eternal life, He gave us something unique, a life different from the one we were living before our calling. At that time, we were "dead" in trespasses and sins because we were living a life that produced death. Those still in the world continue to live this kind of life by nature. Do they have fellowship with God? Do they walk with Him as friends because they agree with Him about how to live and to remain at His side endlessly?
Understanding this fellowship aspect is important to understanding eternal life. "Can two walk together, unless they are agreed?" (Amos 3:3). God created humanity for fellowship, and by nature, we seek it out on many levels. Many find it through hobbies, the arts, politics, intellectual pursuits, social organizations, and sports. The greatest and highest form of fellowship, though, is with God and others who share the common desire to live like God always and whom He is transforming to that end. This fellowship is reserved for those whom God summons.
This means an independent Christian is a contradiction in terms. The Christian church is a community fellowship comparable to a body of which Christ is the Head (I Corinthians 12:12-31). Hebrews 10:25 commands us not to forsake the assembling of ourselves together. Fellowship is not just friendly conversation and geniality. The Bible's writers show plainly it is a tightly knit relationship marked by self-sacrificial love manifested in mutual service, concern, prayer, labor, and helpfulness.