Got to agree workouts like that are really hard!
I got the idea from a BBC program on fitness, I think it was a Horizon's where they had the host doing these sprints and they equaled much longer endurance workouts. I was dubious at first but the results seem to hold up.
https://www.google.com/webhp?ion=1&espv=2&ie=UTF-8#q=what%20...
"quickness" is absolutely about the core muscles. This sort of quickness is not quite what the article was talking about, but for your case it's relevant.
A major reason why squats are so beneficial is because it strengthens your core.
[0] http://myweb.facstaff.wwu.edu/chalmers/PDFs/Triple%20Extensi...
No one here said your core muscles were the only muscles involved in these exercises, but they are critical.
Stop being a jackass.
I'm actually about to start doing squats which are something I've never done much of because I've been lucky genetically and have always had very large calves and thighs. I picked up a book recently espousing the benefits of squats in particular (Starting Strength for those interested) and I think that's the next thing for me to keep or maybe even increase my speed.
The other thing I started doing about a year back now is a upright bike 'sprint' where I basically peddle easy for around 20 seconds, then 10 seconds of full-on max aerobic type sprint. I then rest another 20 seconds peddling easy, followed by 10 more seconds, and finish the same on a final third sprint. When I say full on sprint I mean to the point where I nearly barf every time I complete this part of my workout; you really have to push yourself and not just go a little faster. Prior to switching to this sprint type workout I did a rotation of long day/short days on an elliptical; on my long day I did 40m and on my short I'd do 20m.
What I noticed as I've gotten older is that my endurance remained fairly consistent but my 'explosiveness' was going away. Just switching to the sprints has helped me regain a lot of my speed. In fact a good friend that I've skated with on and off for about 15 years remarked that he thought I was faster now than I've ever been. Considering I've been chasing down 20-something Junior players on break-aways, it seems about right to me.
I think adding in the squats will further help diminish my leg muscle decline and hopefully give me a bit more quickness my first three 'chops' from a dead stop.
I write about all this stuff on my personal blog which focuses on fitness, health and most of all hockey such as this recent post: https://existentialquandary.wordpress.com/2015/09/07/sunday-...
Edit: Book title was wrong - link is in another comment - thanks for linking to it!