You need to disconnect at least the yellow wire, so you don't have both processors trying to talk to it.
De-solder the yellow jumper that goes to pin 24. Connect a jumper from that to your Arduino. Connect a jumper from GND on the Arduino to GND on the Monochron. Then you should be able to power up both systems and see if you can read the sensor with the Arduino.
It works when connected to the Arduino. However, I reconnected it to the monochron and tried using the make-shift oscilloscope again, and this time it didn't flicker at all. I think before it may have flickered because of a bad contact with the pin. What does it mean that it doesn't flicker?