Arduino Yun RFID-RC522 module not working
According to the link you provided, we can see that the RFID-RC522 uses SPI to communicate with Arduino.

On Arduino UNO, the SPI pins are located as follows:


pin 11: MOSI,
pin 12: MISO, 
pin 13: SCK


On the Arduino YUN, this is different though; as stated there (bold highlight is mine):


  SPI: on the ICSP header. These pins support SPI communication using
  the SPI library. Note that the SPI pins are not connected to any of
  the digital I/O pins as they are on the Uno, They are only available
  on the ICSP connector. This means that if you have a shield that uses
  SPI, but does NOT have a 6-pin ICSP connector that connects to the
  Yún's 6-pin ICSP header, the shield will not work.


Hence you can not wire the RFID-RC522 module the same on UNO and Yun.
For the Yun, you'll have to follow this wiring picture:



Pin 1 in image above is the pin that is marked with a dot on the Yun PCB.

Also see the ISP-header layout, at the lower-right corner of the yun_by_pighi Yun pinout diagram.