You have awakened the researcher in me.
One important factor is that water reaches its maximum density when it's at 4 degrees Celsius, which is above its natural freezing point at standard pressure. When you look at a frozen puddle when the air is at around 0 degrees Celcius, you'll see that there's a skin of ice on top of some cold but still liquid water. This is because as water reaches an average temperature 4 degrees or below, thanks to its energy distribution, a layer of water is able to float on top of the warmer, denser water below causing the top layer to freeze (in the right conditions) while the lower layers remain fluid - and the same is true for rivers
[1]. For still water to freeze solid, the average temperature throughout must be below freezing, which is quite a difficult thing to achieve if you look at how energy in substances is distributed.
[2]For flowing water, this gets even worse as there is energy being imparted into the system by the fact that it is flowing. The water is losing gravitational potential energy as it moves closer to sea level and so the water is constantly, though slowly, being fed enough kinetic energy to counteract the friction caused by the water molecules bouncing off each other and the solid edges of whatever they're flowing along. Said friction is, in effect, a small but significant source of heat
[3]. Also, given that the water is not pure (distilled) there will be molecules dissolved in it which will further decrease its boiling point at least a little
[4].
So the answer is.... Yes they can freeze but it's not as easy as making icecubes.
The areas where rivers
would freeze are typically called the Polar Ice sheets which are made mostly of frozen freshwater. So yes, you're basically describing ice caps. I really should have just said that at the start and foregone all of that complicated rubbish.
