Based on a quick google images search, it looks like a Jackson Soloist with three single-coils and a Kahler trem.
Not sure of the vintage but I think the USA Jackson's in 86-88 or so were all custom ordered so they could be anything. I had an 86 Soloist back in the day and the single coils didn't sound very good compared to pretty much any Fender. My humbucker was a Duncan but I never pulled the pickups so I'm not sure if the singles were the generic Jackson's (my guess) or possibly Duncan's (probably not).
My guess is unless you find an interview, you won't be sure as they would put whatever you wanted in them back then.