Romans 1:16 "For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek.:Paul was not ashamed to preach to the Jew nor the Greek (Gentile). He was not ashamed of the gospel, or "good news" about Christ. "News" implies "truth" because, unlike today, "spin" or deception was not "news"! Paul was called and was proud to spread the truth about Jesus!
"Ashamed" is basically "not wanting to be shamed". It is feeling guilty because of a ones own beliefs or associations. The gospel of Christ is about beliefs and associations. Paul didn't feel guilty about either.
First off, there are two types of guilt. True guilt is when the Holy Spirit convicts a person of sin. It's a sinking depressed feeling caused by a behavior or thought. God would NEVER cause a person to feel guilty for teaching the gospel of Christ because gospel implies truth.
The other type is false guilt. It comes from Satan and is directed toward diminishing Jesus. For many people they are ashamed of the gospel of Christ! Satan makes them feel that way.
Being "ashamed" implies that people care about what others think about them. In effect Paul was saying "I'm going to preach to you about Jesus and I don't care what you think about me!" Truly, if Jesus is his Lord, then it's his Lord that he should care about most! Paul cared about what Jesus thought, not about people's thoughts.
When I was born-again I became a "secret Christian" for awhile. I didn't go to work and announced "I got saved!" as I should have. I was embarrassed, not of Jesus, but that others would ridicule me for getting religion. People do that because they are insecure they ridicule and undermine others. Satan uses ridicule to undermine Jesus.
Right now is election time. Many conservative Republicans favor Donald over Hillary. They remain quiet. Why? Because if they speak openly about Trump they will be ridiculed. Trump has many imperfections, but many many less than his opponent. Jesus had no imperfections, but many are still ashamed to speak openly for Jesus in fear of being ridiculed!
"Secret Christianity" is impossible.
The assumption here is that most people can speak. That's a valid assumption. It follows then that if you believe in your heart that God resurrected Jesus, you will tell others about this miracle! When we speak openly it is with confidence (faith) that we speak. It's about associations (God and Jesus) and beliefs (that Jesus is God and that he still lives).Romans 10:9 "That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved.10 For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation."
People can't be secret Christians and "confess with their mouth"! Although salvation is by grace alone even righteous things are DONE because of God's grace in everything that we DO is because God allows it. It's essential that to be born again that people "believe in their heart" and go one step further: tell Jesus and others of your change in heart. You tell Jesus because he likes to hear it and you tell others because you now have hope and want everyone to have the same hope!
With the new Christian, called "babes in Christ" and some mature Christians, there is a battle going on inside. It's a battle between hope and shame. It feels good to know that someday you will be resurrected and live forever, but Christians are attacked viciously by Satan, especially at the time one first believes. Satan says to them "People will think that you are crazy and make fun of you!" Actually Satan is telling a partial truth. Some will, but most won't!
If we look at the Jew and Greek of Paul's time, the Jews were the established religion in that area. Almost all Jews believed in God, but ridiculed Jesus.
If we look at the Greeks, they accepted many gods, but not God. Paul spoke to the Epicurean philosophers who lived for pleasure. They would ridicule because Christians are taught to give up the pleasure for Christ's sake!
In this day and age some are ashamed to admit that they are born-again because most mainline religions have no concept of a new birth. Such religious people are like the Jews who ridiculed because they are in the dark. On the other hand the world ridicules Christians because we cease doing things that they deem pleasurable. As an example almost everyone today drinks alcohol to be sociable and to release inhibitions. I don't drink alcoholic beverages. I've been ridiculed when others say things like "That's sad!" or "It's stupid!" They are the Greeks of today!
Paul was not ashamed of Jesus, his resurrection, nor his own belief in Christ. Satan didn't intimidate Paul! As such Paul was the most vocal spokesman ever for Jesus Christ although he had previously been the most zealous persecutor of Christians! Guess what? We are called to be apostles just like Paul and that can't be done with tight lips. We cannot be secret Christians or Christians who fail to profess publically the name of Jesus!
There is no place for "ashamed" when we should be proud that Jesus saves! If people are truly born-again, they want to tell everyone about their new life. Being ashamed is looking back to what was and that's why Lot's wife was turned into a pillar of salt. God's new creation should be thankful and joyful. Telling everyone about Jesus and your desire that they have the same hope is a demonstration of strong belief. Never ever be ashamed of the One who died for you!