"You should be ashamed of yourself." "I would be ashamed if I were you." "Shame on you!" Most of us have had occasion to hear words like these at some time in our lives. We probably deserved the implied accusations most times. That's just the way human nature is.
In Romans 1:16, the apostle Paul declared, "For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to everyone that believeth; to the Jew first, and also to the Greek [Gentile]." The gospel, "the good news", of Christ centers upon His eternal Sonship, incarnation, sinless life, atoning death upon the cross, and His resurrection from the dead. It is good news to us because His death was payment for our sins. It is good news to us because His resurrection assures us that God was satisfied with the payment.
The gospel is good news because it enables us to tap "the power of God" to effect our salvation from sin. Only the power of God could save us and only the blood of Christ is sufficient grounds for our salvation. That power is available to all, "to everyone that believeth; to the Jew first, and also to the Greek [Gentile]." Neither Jewish religion, nor Gentile religion, nor pagan religion can tap that power; only personal faith in all that God reveals about His Son and His atoning work can save the sinner.
The gospel works. It is the only thing that does work to bring salvation. No wonder Paul was not ashamed of the gospel. Neither should we be ashamed of it nor of the Person and work of which it speaks. Believe on, fully trust in, the Lord Jesus Christ and it will work for you "unto salvation."