I should have phrased that differently. The lungs don't shrink because they're filled with high-pressure air, but shrinking wouldn't damage them. "Crushed" implies damage and it was the wrong word to use. Lungs are soft and effectively can't be crushed, the way that a balloon can't be popped by deflating it.

The guy with the world record for free-diving had the air in his lungs squeezed to a twentieth of its original volume, but lungs are built for that sort of thing. Simply going from inhaling as much as you can to exhaling as much as you can reduces the volume of air in your lungs to a fifth of its original volume, a much bigger absolute change.

(He still had to practice and prepare for years, and he was probably born with an exceptional natural aptitude. Don't try this at home!)

Your lungs can compress to equalize the pressure as you go deeper and expand as you come back up. As long as you start with ambient pressure air in your lungs you won't have issues.

The problem is breathing against the external pressure, you need gas pressure to help expand your lungs again after you exhale. The regulator keeps the air pressure equal to the external water pressure so breathing feels the same no matter how deep you go. With an open loop system you use air faster with depth because each breath is higher pressure and gets wasted when you exhale.

(Lungs withstand being squeezed much better than being stretched so that's why free-divers can hold their breath while swimming down.)

Last sentence of the second paragraph mentions free diver lungs.