Why do you want to tear down the existing plasterboard?
Assuming it's necessary to do that, I would proceed as follows:
1. Tear down existing plaster board (and lathe if it's there).
2. Fasten 2x2 furring strips directly into the brick mortar joints with Tapcon screws. That way, in future, someone can take down this new wall, and pack new mortar into those old holes for an invisible repair if they want to. However, if the existing firring strips are anchored directly to the brick, then the option of restoring the wall to it's "as built" condition is out the window anyway. HORIZONTAL mortar joints should be full of mortar over the entire width of the brick, whereas vertical mortar joints will just have a bit of mortar near the exterior surface of the wall so that the brickwork looks good. So, if you anchor into your mortar joints, anchor into the horizontal mortar joints.
3. Insulate between the furring with foam sheet insulation, preferably the EXTRUDED foam insulation like Roofmate. That's cuz extruded polystyrene foam insulation is impermeable to air infiltration. Expanded polystyrene foam insulation (the white "bead" stuff) is permeable to air and humidity because the spaces inside the foam beads are often interconnected. After fitting the extruded foam insulation between the 2X2's, then "caulk" any gaps around the perimeter of the foam panels with expanding foam insulation. When the expanding foam insulation is fully cured, cut it off flush with the surface of the 2X2's with a hand saw.
4. Hang Drywall on furring. You shouldn't put a vapour barrier between the insulation and the drywall in this case because the extruded polystyrene insulation is impermeable to air migration through it, and therefore acts as it's own vapour barrier. The reason for having a vapour barrier is to prevent air from wafting through fiberglass insulation and the moisture in that air condensing into water as it does. The water will get the insulation wet, causing it to loose it's insulating ability, AND since insulation works by keeping air stagnant, that water will take a long time to dry out. And, of course, keeping the insulation wet for a long time can lead to the 2X2 strapping starting to rot if it's in contact with that wet insulation.
Specifically: Do I screw into mortar joints or into the bricks themselves when attaching the furring? I'd screw into the horizontal mortar joints just so that I always have the option of restoring the wall back to it's original "as built" condition. But, no one's going to bark at you for drilling into the bricks. I think the Tapcons would probably hold better in the stronger brick, tho. But, on the other hand, no one's going to be pulling on the drywall, so it really shouldn't matter.
Do I need a vapour barrier between the drywall and the furring? Only if you use a permeable insulation like fiberglass. Extruded polystyrene is impermeable to air infiltration and therefore doesn't need a vapour barrier. I'd be inclined to use a vapour barrier if you use EXPANDED polystyrene insulation (the white "bead" stuff).
Will this method provide enough airflow around the brick to keep it healthy?
Brickwork can tolerate moisture, and it can tolerate freezing temperatures. It just can't tolerate BOTH excessive moisture AND freezing temperatures because the moisture inside the brick expands as it freezes, and the usual result of that is the face of the brick breaking off in "chips"; a condition called "spalling". If the opposite side of that wall is exposed to the elements, then you should take care to keep the mortar joints on that wall in good repair. If you ever decide to paint the bricks that are exposed to the elements, be sure to use a "Masonary Paint". A masonary paint is a latex paint that will allow humidity to pass through the paint, but not liquid water. Thus, any moisture that does get into the wall (such as air infiltration at a duplex receptacle or light switch) and that condenses into water, can evaporate through the two wythes of brick and through the masonary paint, but any rain falling on the masonary paint won't go through the paint into the wall.
you don't need to know the rest:
The reason why masonary paint can do that is that if you consider a polymethyl methacrylate (Plexiglas) molecule to be a length of wire, then you can imagine a latex paint resin to be that wire scrunched up into a small ball. Unless you can squeeze that wire into a ball with near infinite force, there will be spaces between the parts of the wire in that ball. Masonary paints are made from latex resins where those spaces are larger than the diameter of a single H2O molecule, but smaller than the distance BETWEEN two H2O molecules in liquid water. So, an individual H2O molecule can pass through the paint relatively easily, but liquid water cannot. That is, only HUMIDITY can pass through the paint in both directions, but water cannot pass through it in either direction. Thus, masonary paints allow liquid water inside masonary walls to evaporate out of the wall, but they prevent liquid water from outside from getting into the masonary wall.
