Should I extend existing furring strips or remove them and start fresh?
Those 2x2s are fine as furring strips over block. They are really more a part of the wall finish. The block is the structural wall, it can stand alone without the 2x2s and drywall.  

If you are going to mount cabinets on these furring strips, I think I'd top the furring strips with 3/4" plywood.  

If you want a deeper wall for insulation, plumbing, etc., @manassehkatz mentions it would be smarter to sister 2x4's side by side with the 2x2's than to extend the 2x2s or try to attach 2x4's on edge.  However it may be even easier and stronger to just remove the 2x2's and build a full 2x4 wall with top and sole plates rather than just furring strips.