Can a bad brake light switch cause gear shifter to not move

Hello, Yes, yes it can. It is part of the brake-shift-interlock system which prevents the shifter from moving out of part until you press the brakes. It could also be a bad shift interlock solenoid, ignition lock cylinder, or damaged wires. If your brake lights are not coming on, fix that first. If they are, test the brake light switch, it might have two circuits.

'IF' you mean you cannot move the shift lever at all then yes it is possibly the brake light switch .. However since you stated “will not start” and 'IF' the shift lever will move but will not engage any gear - the shift CABLE has became dislodged from the transmission shift linkage - under vehicle - on drivers side of transmission.