Water always tries to spread out as evenly as possible. I should know, I just flooded a fort that was doing quite well.
Any water column whose surface is considered above ground (aka "Outdoors") will freeze solid, all the way to the bottom, if it gets cold enough outside. This will prevent a surface well from functioning in winter. Water flowing through a tile that at any point has seen the sky (i.e., is considered above ground) will freeze, even if it is now covered over with constructed walls or floors. Water in a tile that is considered "underground" (loo[k] at the tile to see this, by the way) will not freeze, no matter how thin the roof/floor over it is. A constructed floor or wall does
not keep a tile marked underground, even if the natural ground was dug away after the space above was walled/floored over.
Myself, I dig out an 6 to 10 z-level 3x3 underground cistern to hold the water. A single tunnel feeds the cistern through a ramp dug up into the underside of the river/stream/brook. Floodgates are used to control the water so I can drain the cistern if another idiot dwarf dodges into it while sparring. One lever tied to a lockout floodgate normally left open, a second tied to the floodgate controlled by the pressure plate that closes the gate when the water level reaches "full" (triggers the gate open when water drops to 0-4/7 at that level, usually two below the top of the cistern). A third lever at the bottom opens the drain floodgates so I can recover useful items from the idiot that drown in the well.
All of this is from first hand experience. There may be other caveats to keeping your well from freezing, and in working order, but I haven't run into them yet.