On sams faq it says that adding magnets to a HeNe laser makes it brighter. The details are complex as to exactly why this works but it has something to do with preventing part of the IR spectrum from being properly aligned and amplified which means that there is more energy being spent on visible light than on invisible wavelengths. So I would say it's likely that adding magnets only affects the output of gas lasers because plasma is attracted to and can be contained by a magnetic field.
Yes, magnets do help with he-ne lasers, though I'm unsure with diode lasers. I pretty much doubt it since it all takes place in a confined area and not through a big gas tube lasing medium. I had the pleasure of meeting Sam a few times and he's a heck of a guy. In addition to magnets possibly helping, temperature plays a bigger role in he-ne lasers as well as diode lasers. I saw an experiment Sam was helping a professor at Stony Brook with, and it involved wrapping a he-ne laser with magnet wire, except reversing the turns each way. Basically, the whole thing looked like a solenoid, but the magnetic fields actually cancelled out, and in the end, it was just a heater. This was then wired up to a heater temperature controller circuit he built in order to stabilize the laser.