Temperature effects some, mostly all electronic devices, most are rated for +5 to +95 Celsius, working with the unit out of its operational parameters can effect performance.
However, with solid state memory chips the likelyhood that a system will physically freeze due to temperature is highly unlikely.
By solid state im referring to flash memory chips, LCD screens, and actual physical buttons with metal contacts.
HDD's will freeze in the cold temps, and prematurely wear out the drive
ColdCathods will burn out if they're run too cold
Resistive touchscreens, aswell as capacitive seem to degrade there ability to work when cold, I would assume the membrane would also become brittle and break.
Other factors come into play, like the type of battery and glass/plastic/metal used in the construction...but i don't know much about the specifics of mp3 players.
Personally, id recommend buying 3, 50$ mp3 players...apposed to 1 150$ load all of them up with different music, then insted of having 6 hours of music, you have 4*3, 12 hours of music...and when one explodes, which it will, you'll still have 2 others to use.