Hi and welcome to the site.
The comments so far seem quite helpful. However, if you have to bring the AC operated equipment, Japan uses 100VAC at 50 and 60 Hz. So far, I've seen Tokyo and Osaka with 60 Hz, but some other areas of Japan use 100V at 50 Hz ratings. If you stay in the Osaka, Tokyo general area, you should be fine.
As for the voltage ratings, the AC operated machines would be operating at a near "brown-out" or Undervoltage level. The nebulizer might be more sluggish and the mist may not be as small, or it may take longer. I had seen someone mention that the Hill Rom, which would also be at an undervoltage rating, would just need some adjustments increased to compensate for the low voltage. However, if you already have the settings at max, then it will also feel less "effective" than if it were operating at the rated voltage.
I can only speak for Japan, I don't know what the ratings are for Hong Kong, though.
And, technically, a letter from your doctor would have to be put in Japanese, but if all else fails, look up the phone number to the Embassy in Tokyo (the one in Hong Kong would be a good idea, too!)
One extra point to make that no one else has mentioned, as a U.S. citizen, you have your rights while in your country, but other countries view freedom differently, and Japan is no exception. basically, be safe and follow the rules...and don't bring in any antihistimines, some of those drugs are not allowed to be brought into Japan, they may or may not allow it even for personal use..
Good luck and happy flying! P.S. drink lots of water while on the plane, its a long trip!