I've dealt with my fair share of these little suckers... Baja isn't too bad; I've seen worse. They sold a bunch of them and you can get parts off the internet no problem.
It sounds like you MAY have a clogged/partially blocked fuel filter. Some of these quads have an inline fuel filter that can clog. Take you fuel petcock off the tank, clean it out, and replace the inline fuel filter.
Are you dealing with a 2- or 4- stroke engine? I'm assuming a 4-stroke if its a 150cc (most 2-strokes are 50cc)... Pull the plug wire when the quad won't start and see if you still have spark. If you've got spark, its a fuel problem. If there's no spark, check ALL your connections. Chances are that one is loose and you're losing fire when it just dies out. I bet the loose connection is near your stator plate, the CDI box, or the ignition coil...