GFCI's measure the current flowing out the hot and compare it to the current coming back on the neutral. If the imbalance exceeds the rating it will trip. Basically if any current flows to ground bypassing the neutral then it will shut off the power. Typically 5mA but can vary based on GFCI type and use
Because you state the neutral wire attaches to the breaker lug and the pigtail of the breaker to the neutral buss, hot to the breaker lug. This sounds correct. So it should provide the protection
If you pull the switch out and connect the meter to either terminal and ground what do you get? Then flip the switch and test the terminals again
If the switch is connected to the neutral, with the switch "off" it will have one terminal showing hot while the other not. This will look very much like a standard installation. And when you turn the switch "on" both terminals should show little to no voltage. This would be the opposite of what you would see if the switch was connected to the hot line
If the hot is continuous to the light and the neutral is switched. Voltage will pass right through the light and to the switch and stick there if it is off. This is because there is no load so no voltage drop. Once you turn the neutral attached switch on, it connects the neutral to the light. This creates a low resistance path to a zero reference point. Now there is a voltage drop across the light. So neutral in this position will read zero or nearly so since there may be a slight amount of resistance on the wires.