Shakespeare is a HUGE fan of using swelling bodies of water (floods, tides, overflowing rivers) to talk about political turmoil.
In King John, everyone is always talking about rivers, so let's discuss. When King John's authority is challenged by King Philip of France, John asks if "the current of our right roam on?" (2.1.349) Here, he seems to be using the calm, peaceful flowing of a river as a symbol for the way he wants his kingly authority to play out—going about its business, with nobody stopping it. Of course, King John isn't talking to King Philip at a peaceful moment: remember, these two guys are about to go to war outside the city of Angers.
In the next line, King John flips this image around, describing what happens when the river runs into some blockage or "impediment"—you know, someone who gets in the king's way. When this happens, the river shall "leave his native channel and o'erswell / With course disturbed even thy confining shores" (2.1.351-352). King John is saying that if you block the river, it will just flood the countryside anyway; in other words, "Give me what I want, or it will be much worse for you."
Other characters pick up on this kind of language in the play, too. Later on in Act II, when Hubert suggests putting an end to the war by getting Louis to marry Blanche, he compares this marriage to two "silver currents" (2.1.459) flowing together into one river, peacefully. This image is a pretty neat continuation of the earlier one: if each king is a river, then it makes sense for a marriage-alliance between kings—something that makes them family—to be described as two rivers becoming one.
Finally, we also get some interesting river imagery near the end of the play, when Salisbury is suggesting that he and the other rebels should swear loyalty to the king again. The twist here is that Salisbury describes himself and the other conspirators as a flooding river that must now "stoop low within these bounds we have o'erlooked / And calmly run on in obedience / Even to our ocean, to our great King John" (5.4.56-58).
Here, the image of a river overflowing its banks is used to describe rebellion against the king. Shakespeare uses this kind of imagery in other plays, too, especially when people "rise up" (like a tide or a dangerous flood) against an English king. Check out "Symbols" in Henry IV, Part 1 for more on this.