You're not alone in your experience with the traditional squat. You're not the first to notice that it can hypertrophy the glutes. This, for obvious reasons, makes the movement desirable for some and unnecessary for others. Frank Zane, one of the most aesthetic physiques ever, invented his Leg Blaster device because he noticed the same thing about squats. Obviously, for power lifters and others who care more about strength over aesthetics, this is acceptable.
If you're not doing so already, you need to SMR. Be sure to include the pec minor area as if you are overactive they will pull the scaps forward.
You need to stretch peri and post work out as well.
Pre workout: SMR and static stretch (the stretches should not be aggressive and held under 30 seconds in order to minimize down regulating strength)
Peri workout (ie in between sets): static stretch 10 seconds in a non-aggressive manner.
Post workout: static stretch 30 seconds.
In ALL of your stretches, never go to the point of pain and stop at any sign of joint irritation.
Why don't you tell us your current training program? We can take a look and advise.
Also, what do you do for work? Do you sit hunched over for hours at a time?
And yes, if you incorporate these and the other protocols intelligently, consistently, and with a positive mindset, your overall posture will improve.