What is the proper technique to perform an upward and downward glissando on piano? In particular, how should you position your hand, and is it important which finger(s) you use?
Personally I choose to use my thumb nail (going down at least, sometimes a finger nail going up depending on what's easiest, but always the nail.) This hurts to start with, but before long you'll get used to it.
The "down then up" Glissando in Rhapsody in Blue for instance I find easier with the thumb going down, and a finger on the way up - when it's just upwards, often still the thumb nail on the way up.
Alternatives that don't tend to hurt as much for the first times often involve using the fleshy part of your finger, which can work, but for me always sounds like more of a compromise - the tone seems more varied and soft. Of course, this is all subjective, but usually when glissandos are involved in music it's a brighter, louder tone that's sought after.
I've always used right hand thumb nail for downward glissandi. The nail gives you a crisper sound than the side of RH2. You also don't have to worry about forming a callus. For upward Glissandi, I use the nails of RH 2 & 3. Some people only use 3, and that's fine too. The glissando requires practice and you have to work up to them. Don't do them over and over on day one. You've got to build up skin toughness.