Yes, though it is not as significant as it sounds. Due to the number of generations involved, most of the population of England and around a third of the population of the US are estimated to descend from King John, mainly through his illegitimate daughter Joan who had six children and many grandchildren. These and other descendants were no longer royal, so were more likely to appear in commoners' ancestor trees.
The other relevant fact is that nearly all US presidents have been of Anglo Saxon descent, and typically from well-off backgrounds. Eisenhower was German, Kennedy was Irish and Van Buren (the one president not descended from King John) was Dutch. The other 'non Anglo-Saxon' presidents, the Roosevelts and Obama, were actually of partial English ancestry.
Finally, the good preservation of genealogy records in the US and UK and the obsession of the LDS church with genealogy mean that it is easier to prove a royal line of descent for Anglo Saxon US presidents than for many other people worldwide.
Do you mean as a genetic descendant?
You can assume that virtually every person with British heritage (i.e. many hundreds of millions of people, not just a handful of US presidents) is descended from almost everyone (i.e. not only King John) who lived in Britain that long ago. Check this out:
Follow up question: Is there any one president that is more "royal" than the others based on his lineage?