Squats help you develop maximal force production. The more Force you can put into the ground in the least amount of time the higher your vertical jump is going to be.
So in order to have a higher vertical jump you have to be either a) structurally advantaged for vertical jump (long legs and achilles tendon, along with other factors) or b) able to produce lots of force in a short amount of time (power)
power can be seen as being W/t = (Forcexdistance)/time, or in other words the more force the greater the power, the faster the force is exerted the more power. It is generally very difficult to decrease the time of your jumps, as this is largely genetically predetermined. It can however be slightly improved albeit not by much. The force part of the equation however can be expanded greatly by diligent training
First of all don't bother squatting if you don't have a rack to squat from. it's just never going to work out. Find a gym that has a squat rack. If it doesn't have a squat rack it can't be called a gym. More like a "fitness" center than a gym.
also are you measuring the weight with the barbell included? you don't just add the weight on both sides you also have to include the weight of the barbell. Just saying since the weights you listed seem extremely low for someone your age.
As for increasing your squat and deadlift i'd recommend starting on starting strength or stronglifts 5x5. they're both decent programs that focus a lot on the squat and other general strength exercises. choose stronglifts if you feel like you are a complete beginner. do SS if you feel like you have some experience
also you are not going to get much stronger if you lift for endurance. lift in the 4-6 rep range for 3-5 sets for every exercise. squats, deadlifts and the overhead press are the most important lifts you should be focusing on.
power cleans and snatches can also be extremely helpful for increasing your vertical jump (and other olympic lifts for that matter). did you know that out of all athletes that don't directly train for vertical jump that olympic lifters actually have very high Vertical numbers? this is because they train for maximal amounts of Power, they decrease the time of their lifts and the efficiency by perfecting their form, and they train their squats and they perform heavy presses along with their regular form-work with the olympic lifts in order to develop lots of Strength. put the two and two together and you get quite the explosive athlete.
However, again you will probably need to find a new gym for this too. chances are if they don't havea squat rack they sure as hell aren't going to have bumper plates and a platform.
increased my squat from 60 kg - 118 kg in 3 months, while losing weight, other lifts of course also got stronger in that same time period.