1. Your feet should not be pointing straight ahead while squatting, regardless of stance width. This is by far and away the biggest noob mistake. Most people never hit depth / develop knee pain because they squat with their feet straight forward. You should set up with your feet rotated ~30 degrees outwards from parallel (ie your feet should be forming a V with the vertex behind you).
2. When you are squatting down, you should project your knees in the direction that your feet are pointing. Your knees should be out over your feet at all times. Your knees should not go straight forward.
3. Squat (and really all other powerlifting movements) are full body exercises. Your core should be tight; your back should be tight; etc
4. Don't squat in running shoes. If you must squat in running shoes, use a board as the OP stated. Running shoes tend to have a very squishy sole which is great for running but not so great when you are trying to establish a stable base to balance on and push off of. Try to squat in shoes with a hard sole.
It's similar to when people say "chest up" for squatting which actually has nothing to do with your chest directly and is instead a diagnostic for if a person's back and core are tight. If your back and core are tight then your back won't round and therefore your chest will appear "up".
If it hurts excessively don't do it, or proceed with extreme caution. It's probably not a big issue otherwise.
1) Heels should be firmly planted on the ground. This can seem impossible: many people will feel like they can't even get to parallel, let alone ass to ground. My favored solution to this is adding elevation to the heels. A plank or two can add an inch or more to heel height. This gives your foot the support it needs in a squat and helps form good habits. When you rise from the squat, most of the force should be being applied through the heel. Over time you will decrease the heel elevation.
2) Body weight distribution can add to the difficulty of it if you're "back heavy." For my body fat distribution, losing weight enabled me to balance over my heels, while before I would fall backwards.
3) Practice every day, even if it's just a set or two of squats.
4) Hip flexion and ankle dorsiflexion are the key components to a squat. I find the former easier to improve than the latter. The supplementary exercises I've found most useful are Cossack squats, horse stance, and training my pancake split. For the pancake, you want all the folding to come from the hips and not from the lower or upper back. This transfers well to working on squats, because folding from the hips instead of the back brings your center of mass forward more effectively.