I've been trying to get more explosive starts and stops, and faster foot speed which hopefully translates into "on ice speed" some of my routine includes
Jumping from flat ground to an elevated surface (ledge or step up bench).
ill do a mile run at half speed and then walk about .10 of a mile, then run full speed, walk .10 of a mile, run full speed (Repeat)
Setting up two cones a distance apart and sprinting from one cone to another repeatedly.
Eating Healthy/Losing Weight ( I noticed this helped dramatically, Last Winter season i was at 230, i dropped to 210 and noticed my skating ability dramatically improved and I was winning battles), I gained back about 20 pounds in Summer and noticed I slowed down, so this winter I lost about 20 pounds again and went down to 210 and I feel I'm again skating faster.
Get up on your skates at starts, sort of like your running, I kind of run for like 3 steps and then go flat on my blades it gets me started quicker. (This is an on ice type)
Laura Stamms Power Skating YouTube Videos and Books have great information on increasing foot speed/getting faster and more balanced when skating. they have some good information about off ice training as well.
I understand the need for solid core strength, and im wondering if you can do more crossfit style training.
only reason i say this is because squats can be dangerous, lead to knee pain etc if not done properly, and when you are trying to maximize your time you can end up doing more harm than good because you are trying to squeeze in an hour workout.
Prefacing with me not being an expert in the field, but...
The best way to stimulate muscle growth is by moving heavy weights. Squats allow you to put nearly your entire body under load involving lots of stabilizing muscles (i.e. your core). That's why it's so much better than the leg press for instance.
Burpees, lunges, etc are better than nothing though. But I don't think you'll see the same success.
I did kettlebell and bodyweight exercises on and off for years and saw no improvement in my speed and endurance. I did "heavy" (for me) squats and deadlifts for a couple months and saw a jump in my on-ice speed. N=1 but my experience. Again, I'm pretty weak compared to most lifters (definitely putting up beginner numbers) but 3-6 months in the gym could bring me to a very respectable output, probably add 10 pounds of muscle, and double my strength. All good things on ice.
I'm not an expert lifter and am still relatively light but I have had some knee pain issues. I know it's due to technique. I got some good relief doing box squats, then had an on-ice injury and illness that took me out for a few weeks, so just started up last week with a lighter weight. I will likely build a box to use sometime soon if the knee pain lingers. The box gets you in the mindset of sitting back rather than leaning forward (which causes the knee pain), as well as giving you the ability to rest at the bottom of the squat and then explode upward, giving an explosive power component to the back squat (which is a good thing for hockey).
Other interesting options would be hack squats or a trap bar deadlift. I've been looking for one of those for the last couple weeks. You won't get the same benefit as the back squat as you're not stabilizing the load, but you'll get some benefit in terms of working the quads, glutes, and hams, plus some grip strength to boot. Again, better than nothing.
Deadlifts are awesome though. I've been doing sumo lately. Really hits the hams and less stress on the low back. Lots of ab work needed for stability.