Did John of the disciples deny Jesus in order to avoid death?

No. The Bible says it was Peter not John who denied Jesus:

John 13:37,38 "Lord, why can't I follow you now?" asked Peter. "I am ready to die for you!"

Jesus answered, "Are you really ready to die for me? I am telling you the truth: before the rooster crows you will say three times that you do not know me.

verse 37 - Typically over enthusiastic, Peter expressed willingness to die for the Lord. He thought he could endure martyrdom by his own strength.

verse 38 - Jesus checks his "zeal without knowledge" by telling Peter something he himself did not know, that is, before the night was ended, he would deny the Lord three times. Thus Peter was reminded of his weakness, cowardice, and inability to follow the Lord for even a few hours by his own power.

Answer/

John supported the distressed Virgin Mary as Jesus was being crucified and comforted her as she walked weeping through the streets.

In so doing, John risked charges incurring the death penalty for being seen as Jesus' follower and so was not avoiding possible death - see The Lament of the Virgin.