A poster offers up Joe DeFranco's take from an article, 10 Training Myths Exposed:
Filling your belly with air will also create a larger torso and give you a bigger base of support from which to drive. We want as much tightness and support as we can get from the gross muscles of the spinal errectors, abdominals, and obliques.
Another notes the important differences in the two techniques:
Personally, I feel that "pulling in your belly" is potentially dangerous when squatting. When you pull your belly inward, it tends to flex the spine, a.k.a. round your back. This is the last thing you want to happen when you have a heavy weight on your back! After all, unsupported spinal flexion under a compressive load is one of the most common causes of disk herniation... The correct technique would be to contract your erector spinae (arch your back) and fill your stomach with air by taking a huge breath. Then, hold your breath while forcefully pushing your belly out during the most strenuous phase of the lift (Valsalva maneuver). This technique will not only stabilize your spine by increasing the intra-abdominal pressure, it'll enable you to squat more weight! Remember that both techniques of stabilizing your spine have their place in training. For example, I feel that learning how to activate your transverse abdominis is a valid and valuable technique during the lifting of lighter loads. It's also very important for lower-back rehabilitation. On the other hand, if you're participating in heavy strength training, I'd highly recommend performing the technique I described above.
A supporter of the abs-out method:
By forcing the abdominals out and filling yourself with air you essentially create 'fluid balls' (lower torso) and air balls (in the upper torso) that supports the spine, the bigger the ball the bigger the support I suppose. Think of It biomechanically too, the role of the abs is to provide support for the spine, the further the abs are away from the pivot point, the less effort the abs would have to produce to provide support, as the muscle force acts through a longer moment arm . The Abs in debate is based on the premise of creating your own lifting belt with your abs, if during a max exertion you wear a belt you push against it increasing Intra ab pressure, providing a more stable base etc etc. The abs in argument is to create your own natural belt by activating the TVA and increasing Core stability, as an over reliance on belts decreases it.
A poster suggests that there is really no debate:
The two main points of using the "abs out" style are a) increased intrabdominal volume and then b) compressing this increased volume to create more pressure (which provides the greatest spinal stability). the best way is to fill the torso with as much air as possible and either wear a very tight belt (which will artifically limit volume of the container) or powerfully contract all the muscles of the abdomen to limit the size of the container. This is both the safest and will allow for the most weight to be used.
A third method is suggested:
Just to clear things up, there is really no debate at all between abs in and abs out, but rather a misunderstanding. Paul Chek teaches that one should be activating the TVA under light loads and normal everyday movements to stabilize the lumbar spine (huge generalization). What he has a problem with is when people are using the rectus abdominis as a stabilizer in the same situations. He states clearly in his seminar tapes that the rectus is to be a gross stabilize of the spine. In other words using / flexing your abs or pushing your belly out while lifting heavy loads is O.K. What he does not like is using this mechanism all the time, therefore creating a faulty recruitment pattern in everyday situations. If the TVA becomes weak or is not activating at all you will become much more susceptible to lower back injury.
The way I understand the arguments, there's also a third position, that of abdominal "bracing". Think tightening the torso musclature, locking the sternum to the pelvis, not particularly emphasizing pushing the umbilicus in or out.