Does Exhaust brake activate brake lights?

It certainly not the "standard" for brake light to come on when using engine braking. Some do have the chassis setup that way but I would say the majority do not. Certainly, the industry that uses engine braking more than anything else does not have the brake lights come on when using the engine or transmission brake. Have you ever followed a truck heading down a grade? Ever seen their brake lights come on and stay on for the entire trip down that hill? Strangely enough, even though you don't see their brake lights, you know they are maintaining a slow speed (braking) all the way down and none of us have run into the back of one not realizing they were going slow.