first off my squat and vert always went hand in hand. my workouts are a combo of cf, elitetrack workouts, westside barbell, a bunch of other sites and sources and my own ideas. check these out and if you want a good squat check out westside barbell.
as for box height it varies. parallel for me is about 16 inches, but i got the best results it seemed, as far as vert improvment, when i used an 18" box. the weight i was able to use was actually more than my squat.
my friend started the summer with a three quarter squat with about 350 at a bw of 190. he did what i did and he squatted 495 to close out the summer, with a full squat. as to what helped my friend acheive the increase, it was probably both the box squat and the squat as we used them in short three week cycles. we also used the dynamic box squat, later on in the week in the offseason with about 50% of our maxes, concentrating on bar speed.
as for my squat and how i got it. the first time i stepped under a squat rack i was able to do 315 (probably a three quarter squat looking back at best) but i was also a skinny 17 year old. i train by myself, always have except for the three months in the summer, and i use everything for motivation and i go balls to the wall. also, i always worked on my weakest link, whatever it was, whether it was hamstrings, butt, back, whatever, and made it my strong points. think about it, if every part of your body can squat 500 lbs but your hamstrings can only handle 400 lbs. than you need to work on your hamstrings.
my core is strong from squatting or is it my squat is strong from doing core work. either way i have always pushed my strength levels for my core like i did any other bodypart. all guys go crazy over the benchpress, but what if they put the same effort into their core training, oly lifts, etc.
i dont deadlift often, i usually do stiff legged deadlifts. for these i vary either using a snatch grip for more posterior chain involvement of just regular grip(using a hook grip of course). the weights i use for sld get over 450 lbs for reps. as for a regular deadlift i really dont know.
my squat was just over 400 lbs not too long ago and my vert not much over 30, so you can make huge gains. hardwork, research, clean diet, and listening to your body will all help you get what you are looking for. but my biggets gains probably came from the oly lifts as i never did them before. dont neglect these lifts as the explosivness in these lifts is just what you are looking for in a vert jump. oh yeah and dont forget the track work !!!!
hope this answers your questions and helps you obtain your goals.