"Whosoever therefore shall confess Me before men, him will I confess also before My Father which is in heaven. But whosoever shall deny Me before men, him will I also deny before My Father which is in heaven ... And he that taketh not his cross, and followeth after Me, is not worthy of Me. He that findeth his life shall lose it: and he that loseth his life for My sake shall find it." (Matthew 10:32-33,38-39 KJV)

"He that findeth his life shall lose it"

 
Jesus of Nazareth knew that His destiny on the cross (actually, a fixed upright stake with a portable horizontal beam - see Crossing The T) was coming, and often spoke of it well before it happened because He also knew that His life's purpose was to give Himself as the one and only "Lamb Of God."
Christ also used the term "to carry the cross" figuratively for His followers, not meaning that they should be literally crucified (although many subsequently were, including at least two of the twelve apostles, Peter and Andrew, because crucifixion was the very common method of execution used by the Romans), or to walk around wearing a piece of jewelry in the shape of a cross with a Christ idol on it, but meaning that they were to give their lives as a "living sacrifice" (Romans 12:1 KJV) to the cross, to live in obedience to God so that Christ's sacrifice on the cross for them was not in vain. Christ did not die for sinners; He died for repentant sinners. Unrepentant sinners are headed for the lake of fire. "He that findeth his life shall lose it"
Being a Believer isn't just a state of mind, it's a way of life, The Way to Life. Christ's true people are to "take up their cross" by living their lives according to what Christ taught:

"Whosoever therefore shall confess Me before men, him will I confess also before My Father which is in heaven. But whosoever shall deny Me before men, him will I also deny before My Father which is in heaven ... And he that taketh not his cross, and followeth after Me, is not worthy of Me. He that findeth his life shall lose it: and he that loseth his life for My sake shall find it." (Matthew 10:32-33,38-39 KJV)

"He that findeth his life shall lose it" doesn't mean that Christians should go out and deliberately get themselves killed (although, again, many of Christ's people have been, and will yet be, martyred - see Martyrs to understand the origin of the meaning of the word martyr) but rather that they should "lose their lives" that they would have otherwise lived, seeking common carnal pursuits that will be worthless to them on the Judgment Day (see also When Will You Be Judged?):

"Then said Jesus unto His disciples, If any man will come after Me, let him deny himself, and take up his cross, and follow Me. For whosoever will save his life shall lose it: and whosoever will lose his life for My sake shall find it. For what is a man profited, if he shall gain the whole world, and lose his own soul? Or what shall a man give in exchange for his soul? For the Son of man shall come in the glory of His Father with His angels; and then He shall reward every man according to his works." (Matthew 16:24-27 KJV)

When asked by a repentant man what he should do to find eternal life, the Messiah told the man to "take up the cross, and follow Me." How? "Thou knowest the Commandments, Do not commit adultery, Do not kill, Do not steal, Do not bear false witness, Defraud not, Honour thy father and mother," and to live a life of service to God and man (obeying The Ten Commandments automatically does both - see The First Commandment, The Second Commandment, The Third Commandment, The Fourth Commandment, The Fifth Commandment, The Sixth Commandment, The Seventh Commandment, The Eighth Commandment, The Ninth Commandment and The Tenth Commandment:

"And when He was gone forth into the way, there came one running, and kneeled to Him, and asked Him, Good Master, what shall I do that I may inherit eternal life?" "And Jesus said unto him, Why callest thou Me good? There is none good but one, that is, God. Thou knowest the Commandments, Do not commit adultery, Do not kill, Do not steal, Do not bear false witness, Defraud not, Honour thy father and mother."
"And he answered and said unto Him, Master, all these have I observed from my youth."
"Then Jesus beholding him loved him, and said unto him, One thing thou lackest: go thy way, sell whatsoever thou hast, and give to the poor, and thou shalt have treasure in heaven: and come, take up the cross, and follow Me." (Mark 10:17-21 KJV)