"Protestants assume that the "boast" here is a person can brag before God that they "played a role in saving themself". But what does such a claim really even mean? Is a person going to stand before the Judgment Seat and brag to God's face that God must let them into Heaven because they lived a perfectly obedient life without His help? Not likely. Is a person going to brag that God owed them forgiveness because they lived a life without sin? That's illogical. So what kind of bragging to God's face is even possible? I honestly cannot think of anything, and thus I doubt Paul was suggesting people though they could brag to God's face that they saved themself without His help."
Throughout the four gospels, Jesus Christ repeatedly rebuked the Jews for them looking down at other people whom they perceived as being less faithful to the Law. For example, the Pharisees had inquired as to why Christ ate with people of lower stature (Matthew 9:11). He scolded them for seeking after an outward righteousness (Luke 11:38-39). Jesus even said, "unless your righteousness exceeds that of the Pharisees..." (Matthew 5:20). He said to the chief priests and elders, "Truly I tell you, the tax collectors and the prostitutes are entering the kingdom of God ahead of you." (Matthew 21:1). Jesus addressed a rich young man who wanted to enter heaven on the basis of his faithfulness to the Law (Matthew 19:16-30; Luke 18:18-30). Consider also the parable of the Pharisee and tax collector in Luke 18:9-14.
"In terms of the Biblical testimony, here's where I ultimately make my apologetics case, since this is the only thing Protestants claim to accept - though they don't usually accept what the Bible actually says when they're proven wrong from the Bible. First, the very context of Ephesians 2:9 is verse Ephesians 2:11 (and following), wherein Paul begins speaking of Jews versus Gentiles. Between these two camps, there was very much a boasting problem (Ephesians 2:12-15)."
It is true that there was hostility between Jews and Gentiles. The Jewish people did look down on outsiders as disgusting people. However, "ethnic badges" is not the essence of what Paul means when he speaks of justification by works. The Douay-Rheims Translation renders Ephesians 2:9 in this manner, "Not of works, that no man may glory." The Good News Translation reads as follows, "but God's gift, so that no one can boast about it." What is the gift of God that no one can boast about? The answer to that question is our justification which occurs as a result of His mercy. There is evidence in the New Testament of works righteousness being a problem amongst the Jews. Paul would have confronted such errors.