Can I run #14 armored cable behind baseboards?
NEC 300.4(D) covers the type of installation indicated in the question, and it applies to both AC and MC armored cable.  So you'll need to use a steel guard at least 1/16" thick to protect the cable.  

This tip from Fine Homebuilding illustrates this type of installation.  Rather than backing the molding with steel plating, they use a U-shaped channel to protect the cable, which should be much easier:   



The illustration shows a nonmetallic cable and a plaster finished wall.  You will need more depth than the thickness of the drywall to fit armored cable and steel U-channel.  On your outside walls with the furring strip, you can just cut out the foam board with a knife and the furring strips with an oscillating tool to get the additional depth.  On the interior walls, with the drywall directly on the block, I think you'll have to route a channel on the back of the base molding to make room.  In fact it might be better to use that method on the outside walls, too.  

You have another challenge here:  getting the cable from the baseboard channel to the existing boxes.  This might not be too difficult if you can fish the void in the blocks, but it could be a big can of worms.