I'll address this in two ways, as the final answer really does depend on your goals. Unless otherwise stated, when we are talking about squats, I'm assuming back squats.
Health Reasons for Full Squats
Specifically, one of the main reasons to squat to depth (defined as the crease of the hip parallel or lower than the top of the knee) has to do with joint health. The squat is a normal movement that is innate when we first become mobile. When squats are performed with proper technique and to full depth, we get the following benefits:
- Increased posterior chain involvement, which includes calves, hamstrings, glutes, the full core.
- Increased knee stability due to a good balance of posterior chain and quad involvement.
- Increased or maintained useful range of motion due to the mobility required to hit full depth.
- Increased strength and power generation throughout the full range of motion--important for sports performance.
Squats really are a full body exercise. If your goal is general health, squats are a more efficient mode of training than splitting the work between two different exercises. If your goal is to be more powerful, then full squats help you develop that initial drive better than many other modes of exercise.
Training for Aesthetics
If you are a figure or bodybuilding competitor (or simply training for aesthetics), the only thing that matters is winning the competition. You do need to remain healthy, but the only range of motion required is what you need to get into your poses.
In this context, emphasizing quads by using partial range of motion squats, or simply just using leg presses is completely valid. You will still be judged on your posterior muscle development, but you'll have a different set of exercises for those. What's most important here is that you don't ruin your posture or cause your joints to not be aligned properly. That is where injury occurs.
Some bodybuilders or athletes who are cross training will base the core of there training around big compound movements to build the general strength and size, and then zero in the details with isolation work.
Caveats to Partial Squats
Partial squats do have their uses, but typically beginners or people training for general health don't need them.
- Partials can be used to make the squat harder, by stopping just below your sticking point and reversing direction there. This is desirable when you want to get a stronger squat, but it should used as an supplemental lift.
- Partials can be used to perform overload work, essentially getting your body used to handling heavier weights. This is useful when you are trying to get a stronger squat or a stronger deadlift (partials would be off of blocks). Overloads should be used sparingly as they can cause fatigue that lasts for a few days.
- Partials can provide a false sense of strength. This can hit you by getting crushed by weight you can walk out but can't control when you try to descend.
- Partials might not give the results you want. If your goal is quad development, you need to at least go down to where your quads are first engaged. Otherwise you aren't getting the full benefit of even partial squats.
Other Training Modes
Deadlifts are glorious things. Nothing builds back strength like picking something really heavy up off the ground. However, full squats and deadlifts have a good synergy. Full squats help get the bar off the ground--which for many people is the hardest part. There are different variations of leg and core training that help maintain the joint health benefits of full squats, while emphasizing different muscle groups.
- High bar vs. Low bar. One of these will be more comfortable to you. Low bar puts more training stress on your lower back, hamstrings, and hips. High bar allows a more vertical torso which provides a fairly even training stress on all involved muscle groups.
- Front squats. These emphasize the quads and upper back muscles more as you keep the bar across the deltoid muscles in front of your neck.
- Snatch grip deadlifts. These emphasize glute and upper back muscles, helping to increase lockout power and the ability to get the bar off the ground.
- Jumps (both for height and distance). These help generate force more quickly. Training jumps and squats helps both get better.
- Prowler. Both an excellent conditioning tool, and an exercise that helps you generate starting power from a crouched position.
This list is far from exhaustive, and I haven't touched on pause squats, split squats, hack squats, Romanian deadlifts, stiff leg deadlifts, etc. The reason full squats are used as a cornerstone exercise for many athletes is that they have much better carryover to their sport of choice than most of the alternatives. However, most athletes incorporate different variations of the squats to address what they need. Strength training should both increase the usable strength, and help keep the athlete healthy.
If you aren't a competitive athlete, and your focus is general health, full depth squats provide more training efficiency than the partials plus other assistance exercises needed to hit all the areas the back squat hits--including the core muscles.