jgilbert20:
Then I tried connecting 3" jumper cables directly from the Yun to the 16x2 shield (on 5V, A4, A5, GND). This also didn't work.
I2C isn't on A4 and A5 on the Yún.
http://arduino.cc/en/Main/ArduinoBoardYun:
TWI: 2 (SDA) and 3 (SCL). Support TWI communication using the Wire library.
Looks like it is on Digital 2 and Digital 3.
Shield connections that use the dedicated pins next to AREF will probably work as well.