What is this in my electrical box? Can I attach my ground wire onto this screw? Will it be securely grounded?
That screw holds down the cable clamp, which is anchoring your BX cables into the box.  I have seen it used for grounds but it's not ideal at all.  

Push the wires around in the back of the box and look for a hole that is somewhat smaller than the others.   This may be tapped #10-32, if so it is specifically for a ground screw.   They sell #10-32 ground screws plated green, either with or without integrated pigtails.