• I'd done them for years to prepare for ski season.
  See Skiing Exercises at Home | LIVESTRONG.COM
• Body builders do them to build up legs.   Search for squats for bodybuilders.
  
• Runners use them as part of strength training. (see below)

See:
Dr. Pribut Running Injuries: On Strength Training For Overall Fitness
The 4 Best Strength Training Exercises For Runners - Competitor.com
Why squats improve running technique | Running Technique Tips

The proper way to do squats:
Do squats the proper way. Feet wider than shoulder width with toes pointed out.
When doing squats with weights my PT had me down until I just touched a 18" chair or bench. For my 5-9 frame that left my legs just above parallel.
Keep knees back as far as possible. Keep butt back with weight on your heals and lean forward.
This is the most common way , but there is controversy over the proper way.

There is a lot of controversy on whether or not people should squat at parallel, or below. According to USA Powerlifting, a full squat is one in which the lifter bends the knees and lowers the body until the top surface of the legs at the hip joint are lower than the top of the squat depthknees
Source: Squats: The King of All Exercises - Smoot Fitness

Not squatting to full depth takes a hip dominant movement and places a lot of stress on the knees. When done properly a squat will drive the pelvis back and the knees out. This allows the hips, which are packed with the strongest muscles in your body the room they need to operate and generate the power that they need. The knees are essentially along for the ride.
Source: Are squats bad for the knees? | CrossFit Ironborough | Plymouth Meeting

Though "Keep the knees behind the toes" is a popular cuing strategy when coaching squats, the source of this cue--along with the purpose it serves--is up for debate.

Source: Is There a Right or Wrong Way to Squat? | Movement Masterminds
See also
Moderating the Knees In Versus Knees Out Squat Debate | Breaking Muscle
and 3 Squatting Myths That Refuse To Die | Biomechanics Fitness and Performance
Progression methods:
These increase the level over time.

At two hundred squats they give you guidelines on how to start and progress doing body weight squats.
You do an initial test with no added weight to see your basic level which can range from 19 for someone over 40 in average condition to 49 squats for someone under 30 in excellent condition.

They have a 6 week program.
Week 1 does 5 sets with from 9 to 20 squats for people in excellent condition.
Week 6 starts with 5 sets with up to 76 squats squats per set for someone in excellent condition. It ends with 9 sets with from 30-45 squats per set.

Note: Working with weights has the risk of serious back injury. Don't do this without a trainer.

It is based on a proprietary method developed by a sports physiologist in NJ many years ago. I couldn't find any program like this on the Internet, so you'd have to develop your program with a trainer.

It included two types of progressions:

1. Increasing the weight and reducing the number of repetitions.
   It started with 16% of your body weight doing 13 sets of 13 squats slowly 6 seconds per squat.
   ( 2 minutes per set 1:18 squats 0:42 rest.)
   It went to 7 sets of 7 squats with 50% of your body weight. You can do the same sequence with more weight if your are more fit. Up to up to 7 sets of 7 squats with 1 1/2 times your body weight for elite athletes.
2. Increasing the number of repetitions per set and reducing the number of sets. You can proceed up to 170 continuous squats, still 6 seconds per squat.

1. Your effort is 7 or more (see below)
2. Your heart rate is > 70-80% of your maximum HR
3. You have pain in any part of your body

Rated Perceived Exertion (RPE) Scale or Effort Scale

Links:
Ultimate Leg Workout - Leg Exercises To Build Stronger Wheels
The Body Shop: Know Squat? | Runner's World
Calculating Actual Resistance: Incorporating Percentage Bodyweight into Weight Training Workloads
5 Lesser-Known Reasons Why Squats Are So Good for You - Competitor.com
The Good, Bad, and Ugly About Squats for Female Runners | Breaking Muscle
Squats are a Runner's Best Friend | Beyond the Miles
Strength Training For Runners: How To Do It Right
Is There a Right or Wrong Way to Squat? | Movement Masterminds