Can I switch both positive and negative voltage rails on and off using a Teensy 3.2
You should not switch it from the low side.
The LCD is connected to the micro via data lines. If you put a transistor in the GND connection of the LCD, then it is no longer referenced to GND. If the micro's GPIO output signals, then the LCD won't be able to decode them. If the transistor is off and the micro outputs signals, then the LCD will be powered through the data lines and the ESD protection diodes at the input of its chip. That's not recommended.
You need a high side switch. The simplest is to use a PMOS.
LCD GND should always be connected to microcontroller GND.
Before turning off the LCD power, make sure all micro's pins that are connected to the LCD are configured either as inputs without pullup, or as outputs with logic level 0, because when the LCD is not powered it should not receive any voltage on its input pins.