The mass of the submarine increases due to pumping water into the ballast tanks. The weight of the submarine remains proportional to the mass, and varies a tiny amount according to depth (deeper = heavier). The net downward force upon it increases, due to reduced upward force from buoyancy, which is dependent upon density, which is proportional to gas pressure.

The author was correct, but for the wrong reasons.

The weight of the submarine increased due to the additional mass of water in the ballast tanks. The weight of the submarine increased due to the increase in Earth's gravitation at points nearer the center of mass. The apparent weight increased due to a reduction in buoyancy.

But in no way can you increase the weight of something by squeezing it. (Aside from invented hypotheticals like two gas giant planets wrapped in unbreachable envelopes, lightly touching, without rotation.)