Do you think running could help improve your walking?
The trouble is that you gotta be pretty good at walking before you can run.
(actually maybe the analogy doesn't quite work as toddlers tend to do this running/falling combo well before they learn the muscular control to walk - but maybe the analogy really does work very well...)
I'm not sure what you mean by speed squat. Do you mean a fast descent into the bottom position or a fast ascent out of the bottom position or do you mean both?
In Olympic LIfting you need to basically relax everything (excepting your arms) to free fall yourself down under the bar - to hit a tight bottom position under the weight. Hitting that tight bottom position under the weight is important because if you stay relaxed then the bar will come down on you a little and you will absorb the force with your lumbar spine and hip joints instead of your muscles. When people get really good they can drop themselves faster than a free-fall by actively using their arms to push up against the bar which speeds their descent rather than slowing the descent of the bar.
Most people say not to train your squats like that. Don't free fall / dive bomb into the bottom position. There is a risk of injury if you don't hit that nice tight bottom position. So do a controlled descent (kinda like how you do / should do a controlled descent on the bench where you actively row the bar towards you).
With respect to the ascent out of the hole... The idea is to power that weight up as fast as you can (with good muscular recruitment patterns). I'm still getting the hang of getting the muscles to fire at the right time. That means that if I'm really focusing on powering it out the hole I'll get a sticking point where it takes a bit for the relevant muscles to fire. I can get the bar moving much faster out the hole if I 'cheat' my position slightly by leaning forward a little and using my quads to pull it rather than my glutes to push it. But there isn't anything I can gain from doing faster squats cheating the movement compared to slower squats without cheating the movement. Of course the bar moves much faster in the former way of doing things... But I'm trying to move it just as fast in each case.
Hope that helps.