After installing AC or BX Is a Bonding Jumper Required when joining a Receptacle to a metal box?

BX is Trade slang for AC or Armored Cable.  The Internal Bonding strip made of cu (copper) or al (aluminum) should remain in contact with its flexible metal armor along its entire length.  This Bonding strip is used to hold the anti-short bushing or redhead (trade slang) from falling off and is used to protect any size wires from abrasion and Not just #4 awg and larger.  The armor of AC itself is suitable as an equipment grounding conductor as per 250.118 or 320.108.  However when entering boxes where a change is made in order to supply a receptacle, the receptacle must have a bonding jumper in case of short circuit.  The Bonding jumper would be installed from the green screw of the recepacle to the grounded box through the connection of the AC cable to the Box. 

You can find grounding receptacle requirements in article 406 or 250.146.  Basically if the receptacle is attached to a removable cover-mounted receptacle cover , a bonding jumper from the receptacle to the metal box is mandatory.  If a receptacle is attached directly to the box with at least one washer removed and direct metal to metal contact is established, then NO bonding jumper is required.  However certain electrical inspectors still require it.  This is a very debatable topic.  I suggest enrolling in a code change classes and becoming a member of in the IAEI organizations to discuss further.

A very important article is 250.148 which discusses continuity and attachment of equipment grounding conductors to boxes.  Remember the Armor of AC is OK to be used as an equipment grounidng conductor (see 250.148(c) in the NEC or National Electrical Code Book).  Note if a non-wire type equipment grounding conductor is used, grounding the metal box is not required provided the raceway or armor is suitable except when a bonding jumper or grounding continuity is required.