No rear lights

Older wiring diagrams are hard for me to read. In the diagram, the brown wire is voltage for tail lamps, the black wire is ground, the other wire is voltage for brake and turn. Brake and turn use same bulb filament, just not at the same time, they are wired through the turn signal switch. Did you check all applicable fuses, bulbs and bulb sockets. I'd have to check those wiring circuits to find out what is going on.

I have checked bulbs and fuses and both are good. I have not checked the all the sockets at this time.

I was wondering, could this problem be either the headlight switch or the brake light switch?

Any fuse you check, you need to check for voltage also, use a voltmeter. Even if the fuse is good, if no voltage, the circuit is dead. Some fuse circuits are hot all the time, some go hot when you turn on the key. The voltage to your brake switch is usually hot all the time, check for voltage. The switch should be activated by the brake pedal arm.