I'm not completely sure that I understand what you're describing - but I'll give it a shot (I'm not a professional, but I do have experience ;-)
If you can fit a reciprocating saw blade between the cabinet and the cement board, you can run a metal cutting blade along the gap and cut the screws. (Hopefully, they didn't use liquid nails to attach the cement board to the cabinet.) A reciprocating saw can be purchased for under $100, which may seem like a lot, but it's an indispensible tool for a DIY homeowner.
Along the backsplash, you may have to do a little bit more destructive demolition - (cut a little deeper, chisel, hammer, prybar...) hopefully the repair work won't be too difficult.
This may chew up the top of the cabinets a little, but a new countertop and/or some trim could cover up any damage.