Why are American prisons packed full of Christians?

Christians combine dishonesty with stupidity. It is not surprising so many of then get caught. 

because secular society is persecuting them

America is a so-called majority Christian country...even though most are nominal at best.  Also many people convert in prison because there are a number of very fine prison ministries. 

Matthew 25

 “But when the Son of Man comes in His glory, and all the angels with Him, then He will sit on His glorious throne.  And all the nations will be gathered before Him; and He will separate them from one another, just as the shepherd separates the sheep from the goats;  and He will put the sheep on His right, but the goats on the left.

 “Then the King will say to those on His right, ‘Come, you who are blessed of My Father, inherit the kingdom prepared for you from the foundation of the world.  For I was hungry, and you gave Me something to eat; I was thirsty, and you gave Me something to drink; I was a stranger, and you invited Me in;  naked, and you clothed Me; I was sick, and you visited Me; I was in prison, and you came to Me.’ 

 Then the righteous will answer Him, ‘Lord, when did we see You hungry, and feed You, or thirsty, and give You something to drink?  And when did we see You as a stranger, and invite You in, or naked, and clothe You?  And when did we see You sick, or in prison, and come to You?’  And the King will answer and say to them, ‘Truly I say to you, to the extent that you did it for one of the least of these brothers or sisters of Mine, you did it for Me.’

 “Then He will also say to those on His left, ‘Depart from Me, you accursed people, into the eternal fire which has been prepared for the devil and his angels;  for I was hungry, and you gave Me nothing to eat; I was thirsty, and you gave Me nothing to drink;  I was a stranger, and you did not invite Me in; naked, and you did not clothe Me; sick, and in prison, and you did not visit Me.’  Then they themselves also will answer, ‘Lord, when did we see You hungry, or thirsty, or as a stranger, or naked, or sick, or in prison, and did not take care of You?’ 

Then He will answer them, ‘Truly I say to you, to the extent that you did not do it for one of the least of these, you did not do it for Me, either.’  These will go away into eternal punishment, but the righteous into eternal life.”

"Every person is made in the image of God. No life is beyond His reach. Founded in 1976, Prison Fellowship® exists to serve all those affected by crime and incarceration, and to see lives and communities restored in and out of prison—one transformed life at a time." - https://www.prisonfellowship.org/?mwm_id=383989518...