Well, the documents do not claim that packing is the primary reason of lung squeeze, or that you won't get lung squeeze if you do not pack. They only tell packing may be an important and a frequent factor in many lung squeeze cases, although it sounds paradoxical.
Previously I expressed also another suspicion (though absolutely not supported by any research) that some cases of lung squeeze may surprisingly actually happen during the final stage of the ascent. Imagine a freediver who strongly packs on the surface - the overpressure serves to stuff his/her lungs with some extra air to be used for equalizing, and to allow descending to a greater depth before reaching the negative pressure (hence actually preventing lung squeeze). I see no reason why it should not work for this very purpose. On my mind the risk it much greater at the very beginning of the dive, when the lungs are under the full pressure (higher than the environment) - a sudden move during the duck dive, or the first strong kicks may induce an injury.
When descending just few meters, the risk should be eliminated, because the pressures are now in equilibrium. Now what happens during the dive: both due to the the depth, and due to the increasing diving response, the progressing vasoconstriction moves blood into the veins and capillaries in the lungs, reducing so their inner volume and preventing so the negative pressure in the lungs and avoiding barotrauma in the depth (Blood Shift). So far so good.
Now you start ascending, the environment pressure decreases, and the lungs expand. You'd think you are now safe from the barotrauma, but on my mind the worst moment only comes as you approach the surface - the pressure inside the lungs starts to raise over the level of the environment, not only because of the initial packing, but the overpressure is amplified by the still progressing vasoconstriction, and reduced lung volume. So actually you finish with even higher pressure in the lungs than you started when fully packed. And of course, adding the fact that the blood pressure is much higher at the end of a breath-hold, and that there may be quite strong contractions in this phase of the dive, on my mind it increases the risk of a barotrauma (lung squeeze) very seriously. And especially at those divers who use a nose-clip or closing their epiglottis, or the soft palate+mouth on the ascent - which does not allow the overpressure to leave by the nose or mouth.
The earlier mentioned loss of elasticity of the alveoli tissue due to frequent extreme packing may be another factor that may increase the risk of a barotrauma during the ascent - if the alveoli walls collapse due to the lost elasticity, instead of shrinking, touching so each other, I imagine it can happen that they stay collapsed (sticked together) either during the whole ascent, decreasing so again additionally the lung volume (and increasing the resulting overpressure), or at some point will explosively fill with the high-pressured air from the non-collapsed parts, potentially damaging them.
I repeat that all of this is solely my uneducated hypothesis based only on my reading of posts of others on DB, and the reading of some other sources, and that I have no scientific backing for it. There were also some discussions on DB, where others did not agree with it. I am still persuaded that some (when not many) of lung squeezes actually do not happen in depth due to a negative pressure, but rather due to the packing, either at the moment of the immersion, or in the final phase of the ascent.
That told, I apologize for the lengthy off-topic post in this thread, but since the question was raised, I felt obliged replying in as much details as I could - hopefully helping someone avoiding an injury.