Your lungs operate on the principle of differential pressure. During normal breathing, hen the pressure inside your lungs becomes less than that of the surrounding atmosphere, air rushes in to normalize and fill it until they are again equal pressures. Gradually increasing or decreasing the pressure of the atmosphere around you (as on a plane or while diving) does not change the physical difficulty of breathing. It may slightly alter the diffusion coefficients of gasses passing through the membranes in your lungs, but this effect is mild at the pressures you could attain in an aircraft hull (and toxic at higher pressures! [0]).
For this reason, you'll notice in images of people in iron lungs, their heads are outside of the device. This allows those who cannot create negative pressure (due to damaged or paralyzed diaphragms or ribcage muscles) to follow a different path. The pressure outside of their chest becomes lower than that experienced in their lungs, forcing an expansion; in order to breath out, the pressure in the chamber is increased. In an aircraft hull a person's lungs/trachea/mouth would be exposed to the same pressure as their chests.
This of course does bring up the very valid question: What happened to all of the iron lungs after the decline of Polio?
[0] https://en.wikipedia.org/wiki/Nitrogen_narcosis
EDIT: Oh I realize I didn't fully discuss the possibility of using a plane as a hyperbaric chamber [1]. The constraints around this concept also rule it somewhere outside of what I would consider feasible. Aircraft typically use bottled oxygen (I believe the 737M does) or a chemical generator, neither of which can produce continuous oxygen or fill the entire plane with it to high levels [2]. If you were able to outfit patients with individual tanks (which would also have to accommodate for increased pressure), the gains seem mild at best compared to 100% O2 in a hospital setting. Fick's Law for the membranes in your lungs is roughly
Rate of diffusion = (Area * Solubility of gas * concentration gradient) / (membrane thickness * sqrt(molecular weight of solute))
The factor that increasing pressure would modify is the solubility of the gas, which according to Henry's Law [4] is directly proportional to pressure. At absolute best (100% O2 on the plane at 2atm), you could expect to get 2x improvement in blood oxygen saturation. Unfortunately I don't believe the breathing issues that are being described can be overcome by this strategy. It's an incredibly creative concept though!
[1] https://en.wikipedia.org/wiki/Hyperbaric_medicine
[2] https://en.wikipedia.org/wiki/Emergency_oxygen_system
[3] https://clinicalgate.com/gas-exchange-between-air-and-blood-...
