I would show the digits as part of this
void digitalClockDisplay(){
// digital clock display of the time
Serial.print(hour());
printDigits(minute());
printDigits(second());
Serial.print(" ");
Serial.print(day());
Serial.print(" ");
Serial.print(month());
Serial.print(" ");
Serial.print(year());
Serial.println();
}
Make an array of number fonts, drive the segments from that array.
Every 4 seconds, update a digit. Display second only when button is pressed.