at the e-box, there are 4 wires you need to look for.
a brown & red pair, and a black and white pair. about 4-6" outside the box, each pair will have a plug.
you can unplug the b/w pair and try to start the ski - if it starts, your stop button needs repairing. you'll have to choke the engine to kill it if it starts this way.
you can also use a tester and test the "switch end" of those wires for continuity. Hook a test lead to each wire (reb/brown or black/white) - if you have continuity before pressing a button - it needs repairing. If you press the button and have continuity, it works. If you press the button and it doesn't - it needs work. Repeat for the other button.