Can I also attach a button to those pins?
Not while the GSM shield is using them.
Would it be possible to move the communication with GSM shield to other pins?
Bend the pins so they don't fit in the slots. Solder wires to the pins, and to two other pins, and use the two other pins in the SoftwareSerial instance setup.
My guess it's that GSM needs to use those pins for a certain reason which I don't know.
Laziness.
None of this, though, will allow you to turn off the GSM shield. Without doing that, you won't save power.