Do I have to use armored (MC) cable to connect to a pot (recessed) light fixture?
You are required to use whatever type of wiring your local code permits for this type of connection.  If you are in a residential location then it is likely NMD90; if you are in a commercial location then it will likely require BX/AC. The shielding you are referring to between the can and the box isn't really BX/AC; its just flexible metal conduit. It looks similar to BX but it's not the same thing.

If you do use BX/AC, make sure to use plastic bushings at both ends of the cable so you don't nick the conductors with the sharp BX/AC.

Usually with cans, there is a friction cable clamp you just push the NMD90 through, however if you use BX/AC then you will need to use one of the punch-outs and a proper cable clamp to secure the cable.