An easy way to view this is you are too prone to leaning forward when you come up from a squat. So, how can you practice staying more upright?
- "Stick your chest out"
- "Make it so you can always read the letters on your chest"
- "Chest up!"
- "Look straight ahead"
- "Look up"
There is an art to cueing. What works for some doesn't work for all. However, you can see the general idea above is to get the spine more upright. For instance, the body tends to follow the eyes. If you're looking down, that could cause you to round over some.*
This is the easiest and fastest route to correction, though it's not fail-safe.
Forced Squat Practice
Sometimes voluntary action just isn't enough. We can then change the environment.
Practice squatting facing a wall:
Credit: Diesel Strength
Because of the wall in front of you, you won't be able to lean forward much (at least not without hitting your face). It's a good way to groove the movement and get the muscles you want primed and acclimated. For instance, you could do this as a warm-up, or in between squat sets.
You could take this further and do something similar with a bar on your back. For example, if you set-up your squat so the bar is very close to the rack, you will have the same effect as the wall squat. That is, if you lean forward, the barbell will hit the rack.
This is in essence what a Smith Machine does*:
Front Squatting is another approach. Because the bar is on the front of your body, you will be forced to stay more upright. If you lean forward with the bar on front of you, you'll dump it.
Practice during the day too
Finally, if you say, sit in a rounded upper back posture all day (kyphosis), or you just lean forward staring at a computer too much, that could be worth working on too. After all, if you're practicing leaning forward all day, it's natural you get very good at that motion.
In fact, notice the position above. The person has the hips and knees bent. They are actually in a squat position. They aren't doing it intentionally, but if you sat like a red X above, you'd be practicing being in a squat with a forward lean for hours and hours every day.
More help in this domain.
A muscle by muscle focus can be overrated
Many will approach a problem like this as "you're trying to use your posterior chain more than your anterior."
Which makes sense. If you were to squat without leaning over at all, you're going to feel that more in your quads than your lower back, glutes / hamstrings. After all, this is why a Good Morning is a posterior chain exercise. (Nobody does them because they're trying to hit their quadriceps.)
You could even get fancier and say "Your thoracic extensors aren't working as well as they should." Though that's not necessarily true. You could be leaning forward but not rounding forward. There is a difference. (Rounding forward is where the injury concern comes in. Leaning forward without rounding is fine. Again, that's the point of a Good Morning, and the Deadlift exercise.)
Again, there can be merit here. However, if you instead focus on the movement you're trying to correct, the muscles will take care of themselves. Plus, there is no guarantee if you went on a couple month quad. strengthening program, you'd suddenly start squatting more upright. The body is rather specific in its adaptations. A strength gain can happen in the absence of a technique change. (Just because you take steroids doesn't mean you immediately can hit a baseball further. You need to still practice swinging a bat.)
*I realize some of the die hards here may not be partial to looking up ala Rippetoe fans or those worried about hyperextension of the neck. As well as some might not like the idea of using a Smith Machine. Keep in mind the question is how can the person stay more upright. It's not how can they squat with everything in ideal alignment. For some, a compensation at e.g. the neck can be worth better position at e.g. the lower back.