John 5:28-29 (NKJV)
Do not marvel at this; for the hour is coming in which all who are in the graves will hear His voice and come forth-- those who have done good, to the resurrection of life, and those who have done evil, to the resurrection of condemnation.
I heard my pastor say one time, people who don't believe in God think they can just crawl in the grave and pull the dirt over their eyes and cease to exist. You can't escape that way. There is coming a day of judgment. When the Bible speaks of being saved, the idea should come in you mind: "saved from what?" God's judgment is what you can be saved from. Everyone is guilty of sin and is deserving of the punishment that God's justice requires. He offers a way out, but that offer must be believed and received.
Hebrews 9:27 (NKJV)
And as it is appointed for men to die once, but after this the judgment
John 5:24 (NKJV)
"Most assuredly, I say to you, he who hears My word and believes in Him who sent Me has everlasting life, and shall not come into judgment, but has passed from death into life.
I have some additional thoughts about the idea of eternal existence. In the information sciences, there is a principle that there is a difference between the information and the medium the information is carried on. The medium of the information might be ink and paper or a CD-ROM or even smoke signals in the sky. The medium has mass, or energy as in a radio signal. Anything that has mass or energy must exist in time. The actual information however has no mass. Weigh a blank CD-ROM precisely. Now put a lot of information on it like some music. Weigh it again and there is no difference in the weight. The information has no mass but it is real. Anything that has no mass is timeless; it exists outside of our space-time domain. Your body has mass but the way your body is made is entirely specified in the genes of your DNA. Your body is the hardware but your gene arrangement is software. The specific arrangement of the genes in your DNA is the information that defines you. That information is timeless or shall I say--eternal. You are an eternal being whether you like it or not. You will exist somewhere for eternity!