On my 2008 kawasaki ninja 250, is the brake light supposed to come on when applying...

Absolutely yes!

There should be a brake switch trigger near the back end of the brake pedal - depress the pedal and see what moves when you do that. The trigger will probably be a black cylinder with a wire coming out of it going up the frame. Make sure the brake pedal actually depresses the little piston on this cylinder, and if it does appear to be mechanically working correctly, then check that wire - make sure its connector is plugged in correctly and the wires don't look frayed or cracked. Sometimes unplugging and re-plugging a connector can get it working again (by cleaning off corrosion on the contacts inside).
 
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