I broke the knockouts on the blue receptacle boxes while doing electrical rough in. Now what?
The trap doors serve a few purposes in plastic boxes:


They keep stuff in. Sparks, flames, and other rogue electrical gremlins.
They keep stuff out. Fingers, pets, or anything else capable of getting electrocuted or causing a short.
They act as the cable clamps for non-metallic cables entering the electrical box.


Fire proof foam or caulk would likely satisfy (1) and (2).



The plastic box conundrum.

On the cheaper plastic boxes (you know, those really flimsy ones), even if you don't try to break off the trap doors, they will, out of spite, break off on their own. Even if they don't break off, they typically end up so loose that there is no way they are an effective cable clamping mechanism.

This begs the question: if these boxes are manufactured this way, then shouldn't this be acceptable?

For certain single gang boxes where the cable is secured within 8 inches of the box, this is acceptable; otherwise, it is not acceptable. 


  2014 NEC
  
  314.17 Conductors Entering Boxes, Conduit Bodies, or Fittings. Conductors entering boxes, conduit bodies, or fittings shall be
  protected from abrasion and shall comply with 314.17(A) through (D).
  
  (C) Nonmetallic Boxes and Conduit Bodies. [...] Where nonmetallic-sheathed cable or multiconductor Type UF cable
  is used, the sheath shall extend not less than 6 mm (1/4 in.) inside the
  box and beyond any cable clamp. In all instances, all permitted wiring
  methods shall be secured to the boxes.
  
  Exception: Where nonmetallic-sheathed cable or multiconductor Type UF cable is used with single gang boxes not larger than a nominal
  size 57 mm x 100 mm (2-1/4 in. x 4 in.) mounted in walls or ceilings,
  and where the cable is fastened within 200 mm (8 in.) of the box
  measured along the sheath and where the sheath extends through a
  cable knockout not less than 6 mm (1/4 in.), securing the cable to the box
  shall not be required. Multiple cable entries shall he permitted in a
  single cable knockout opening.




Solutions?

If you do not want to reinstall new boxes, you need to plug any holes that are not occupied by electrical cables. Then, be sure you have secured your cables to the stud or some other suitable surface within 8 inches of the box. Please note this exception does not extend to anything larger than a 1-gang box, so if you do have larger boxes you will need to make a judgement call.

If you do want to reinstall, I recommend using the heavier duty plastic boxes or metal boxes. From experience, the thicker blue boxes made by Carlon* are less susceptible to trap door failure.

*Other manufactures might make this style box as well, but Carlon is the only one I've seen in my area. This is not a Carlon endorsement.