Squatting down to a box likely wont transfer fully to a free movement and the position isn't the same.
The drills I use with beginners with mobility issues are the following. These are done DAILY. Ideally twice on workout days (beginning and end of session).
It's a form of loaded stretching that I call EQI, they are discussed in my second book (Theory and Applications of Modern Strength and Power Methods) and I just finished writing an article about them for T-nation so I don't want to give too much details. But it is the superior way to improve active range of motion (OM while the muscles are being contracted).
The first two exercise are really just to get your body better prepared to the the third one but they have benefits of their own and MUST be done if you can't get down comfortably in a full overhead squat. I will present the drills then give you the way to perform the exercises and the parameters.
DRILL #1 EQI split squat
DRILL #2 EQI push-ups
DRILL #3 EQI in overhead squat position (see picture at top of post)
EQI are not just regular stretches; they are done when the muscles are contracting. You lower yourself to the low position an actively attempt to hold yourself there (isometric action). As fatigue sets in your muscles get weaker and as a result it becomes hard to hold yourself up... you will begin to "sink down" basically reaching a deeper point in the range of motion. YOU ARE STILL ACTIVELY TRYING TO HOLD YOURSELF UP! You just can't because of the fatigue. This results in the muscles being slowly stretched while they are contracting. This will stretch the muscles themselves not just the non-contractile elements like regular stretching does.
I find this method to work faster than anything else BUT you must be able to handle pain! The burn is t least comparable to a monster drop set and the more you can tolerate the more effective the exercise is.
Ideally we shoot for a duration of 60-75 seconds per set for a total duration of 3-5 minutes. At first you might only be able to handle 45 or so seconds (where lactic acids starts to really accumulate) but work your way up to 60-75s. When you reach that time frame you can extend it even further to 90 or so seconds or add some more resistance.
For the overhead squat, go down as low as possible and hold the position. As you will fatigue you will be able to go a bit lower and every set will be deeper.
I had a 40+ years old former swimmer with BAD thoracic mobility and super tight hip flexor able to do a solid full overhead squat and snatch in less than 2 weeks...and he had been trying for 2 YEARS! The guy is a great therapist and knows all about regular mobility drills.