"Not by Paul's sermons, but by his bonds, was the attention of the court attracted to Christianity. It was as a captive that he broke from so many souls the bonds that held them in the slavery of sin. Nor was this all. He declared: “Many of the brethren in the Lord, waxing confident by my bonds, are much more bold to speak the word without fear.” Philippians 1:14 . AA 464.1"
"And many of the brethren in the Lord, waxing confident by my bonds, are much more bold to speak the word without fear."
The EGW passage directly quotes Philippians 1:14, making it a clear and explicit allusion.
"Paul's patience and cheerfulness during his long and unjust imprisonment, his courage and faith, were a continual sermon. His spirit, so unlike the spirit of the world, bore witness that a power higher than that of earth was abiding with him. And by his example, Christians were impelled to greater energy as advocates of the cause from the public labors of which Paul had been withdrawn. In these ways were the apostle's bonds influential, so that when his power and usefulness seemed cut off, and to all appearance he could do the least, then it was that he gathered sheaves for Christ in fields from which he seemed wholly excluded. AA 464.2"
"Let your light so shine before men, that they may see your good works, and glorify your Father which is in heaven."
Both passages emphasize the importance of Christians living in a way that reflects their faith, allowing others to see their good works and glorify God. Paul's example in prison is presented as a way his light shined.