If Jesus finds me, can I lose me?

These things have I written unto you that believe on the name of the Son of God; that ye may know that ye have eternal life, and that ye may believe on the name of the Son of God.

1 John 5:13, KJV

Someone has said that we can become followers of Jesus and then reject Him and so not be able to enjoy heaven and spend eternity separated from God.  If we “have” eternal life right now, as the above verse states, when will “eternal” end?  This person told me that eternal means “timeless” or “without respect to time.”  And I understand this is his explanation that eternal life can end.

I submit that if we have eternal life it doesn’t matter what time it is, we have it.  We have it now, we have it later, we have it much later, we have it without respect to time.

Knowing God, Jesus and the Holy Spirit is eternal life.  If I reject Jesus do I stop being His son?  If my son rejects me (a regular occurrence) does he stop being my son?  If Christ died for me when I was His enemy, what will stop Him now that I am His son?