Translation:American soldiers are not fighting by the rivers.
That's the best accepted translation, actually. Maybe you had a mistake somewhere else in your sentence?
The American soldiers is correct if you are speaking English. The Americans soldiers is wrong. Bad translation.
The american soldiers don't fight at the river should also be accepted. Reported.
I'm still of the firm opinion that -nál shouldn't be translated as "next to".
-nál should be "at" or "by", and mellett should be translated as "next to" or "beside".
What's the difference? In this particular sentence, not much. But that changes once you go to things that are more compact and orderly. Consider a car, for instance. If you say "Az autónál várok", it tells me that you're somewhere near the car. I know where we left the car before we went shopping, and as such you're now telling me where I can find you: I go where we parked the car, and I'll see you there. The car is my reference point to find you.
"Az autó mellett" is more particular. You're not just at the car, but you're next to it. Not in front, not behind, but at its side. You could say "A fekete autó mellett állok" - "I'm parked next to the black car" if someone is looking for your car in a parking lot, since (mostly) the cars are neatly aligned.
Or about buildings in a town: If you say "I live by the theater" ("A színháznál lakom") I know that your residence is somewhere close to the theater, can be across the street, can be a bit futher away, but in the general area. On the other hand if you say "I live next to the theater" ("A színház mellett lakom"), your house is right next to it. The theater is your neighbouring building.