Would a 737 pilot use flaps in nose dive?
No, flaps are never used during a nose dive.

We need to distinguish between two scenarios:


Intentional nose dive: this is actually called a rapid descent maneuver. The FCTM (Flight Crew Training Manual) has a whole section dedicated to it. The general procedure looks like this:


From the Boeing 737 NG FCTM (7.5 Maneuvers - Rapid Descent - Level Change (LVL CHG)):


  Because of airspeed and altitude protection and reduced crew workload, use of the
  autopilot with LVL CHG mode is the recommended technique for rapid descents.
  Use of the V/S mode is not recommended.
  
  Initiate a turn, if required, using HDG SEL. Set a lower altitude in the altitude
  window. Select LVL CHG, close the thrust levers and smoothly extend the
  speedbrakes. Autothrottles should be left engaged. The airplane pitches down
  smoothly while the thrust levers retard to idle. Adjust the speed as needed and
  ensure the altitude window is correctly set for the level off. During descent, the
  IAS/MACH speed window changes from MACH to IAS at approximately 300
  KIAS. Manually reset to VMO as needed.
  
  When approaching the target altitude, ensure the altitude is set in the MCP altitude
  select window. Altitude capture engages automatically. Adjusting the command
  speed to approximately LRC or 300 knots before level-off aids in smoothly
  transitioning to level flight. The pitch mode then controls altitude and the thrust
  levers increase to hold speed. Smoothly return the speedbrake lever to the down
  detent during the level off maneuver.
  
  When descending with the autopilot engaged and the speedbrakes extended at
  speeds near VMO/MMO, the airspeed may momentarily increase to above
  VMO/MMO if the speedbrakes are retracted quickly. To avoid this condition,
  smoothly and slowly retract the speedbrakes to allow the autopilot sufficient time
  to adjust the pitch attitude to maintain the airspeed within limits.
  
  When the speedbrakes are retracted during altitude capture near VMO/MMO, a
  momentary overspeed condition may also occur. This is because the autopilot
  captures the selected altitude smoothly by maintaining a fixed path while the
  thrust is at or near idle. To avoid this condition, it may be necessary to reduce the
  selected speed and/or descent rate before altitude capture or reduce the selected
  speed and delay speedbrake retraction until after level off is complete.


In case a higher descent rate or lower descent airspeed is required, the landing gear can be extended additionally to the speedbrake, but flaps are never mentioned:


  Landing Gear Extended Descent
  
  The rapid descent is normally made with the landing gear up. However, when
  structural integrity is in doubt and airspeed must be limited, extension of the
  landing gear may provide a more satisfactory rate of descent.
  
  If the landing gear is to be used during the descent, comply with the landing gear
  placard speeds.

Accidental nose dive: this would fall under the general term of upset. The recommended procedure is described in the FCTM (7.26 Maneuvers - Upset Recovery):


  Nose Low, Wings Level
  
  In a situation where the airplane pitch attitude is unintentionally more than 10°
  nose low and going lower, the airspeed is increasing rapidly. A pilot would likely
  reduce thrust and extend the speedbrakes. Thrust reduction causes an additional
  nose-down pitching moment. Speedbrake extension causes a nose-up pitching
  moment, an increase in drag, and a decrease in lift for the same angle of attack. At
  airspeeds well above VMO/MMO, the ability to command a nose-up pitch rate
  with elevator may be reduced because of the extreme aerodynamic loads on the
  elevator.
  
  Again, it is necessary to maneuver the airplane's flight path back toward the
  horizon. At moderate pitch attitudes, applying nose-up elevator, reducing thrust,
  and extending speedbrakes, if necessary, changes the pitch attitude to a desired
  range. At extremely low pitch attitudes and high airspeeds (well above
  VMO/MMO), nose-up elevator and nose-up trim may be required to establish a
  nose-up pitch rate.


Also here, flaps are not mentioned, so only speedbrakes should be used to increase drag (and increase the nose-up pitching moment).