Do all conductors on a 20-amp circuit need to be 12 AWG?
Yes, all conductors protected by a 20 ampere circuit breaker must be at least 12 AWG copper. There are a few exceptions and loop holes, but none of them apply to your situation.


  National Electrical Code 2014
  
  Chapter 2 Wiring and Protection
  
  Article 240 Overcurrent Protection
  
  240.4 Protection of Conductors. Conductors, other than
  flexible cords, flexible cables, and fixture wires, shall be
  protected against overcurrent in accordance with their ampacities
  specified in 310.15, unless otherwise permitted or
  required in 240.4(A) through (G).
  
  (D) Small Conductors. Unless specifically permitted in
  240.4(E) or (G), the overcurrent protection shall not exceed
  that required by (D)(1) through (D)(7) after any correction
  factors for ambient temperature and number of conductors
  have been applied.
  
  (5) 12 AWG Copper. 20 amperes


The rationale here, is that 12 AWG copper conductors are rated for 20 amperes, whereas 14 AWG copper conductors are only rated for 15 amperes.  So if you put 20 amperes through a 14 AWG conductor, the conductor could heat to dangerous levels. 

That would make it a terrible idea to use 14 AWG copper conductors, on a circuit protected by a 20 ampere circuit breaker.