Sure, why not: King John lived 800 years ago, so there's been more than enough time for his descendants to blend throughout the British population. Note that the key point in the article is that this claim isn't unique to "presidents of the USA", but is simply another illustration of how everyone is related:

Andrew Millard of the University of Durham calculated the probability that anyone with Anglo-Saxon ancestry descends from King Edward III (1312-1377) to be as high as 99.997%, and conservatively estimated that Edward III has around 100 million descendants in the British Isles, Europe and former British colonies, including the USA, Canada and Australia.

There have many posts in this sub on this question, most popular subjects being Charlemagne (are all Europeans descended?), King William the Conqueror (British?), Genghis Khan (Asians?), but you can basically substitute their name for any other person who lived a long time ago, whether they were a monarch or milkmaid.

Here are a couple of posts for some background:

• an intro on the subject by /u/Georgy_K_Zhukov: How reliable is ancestry.com? Is it based on primary source data? - start at Are You In Fact Descended From the King of Norway.

• an /r/AskScience AMA with a couple of geneticists AskScience AMA: We are the authors of a recent paper on genetic genealogy and relatedness among the people of Europe. Ask us anything about our paper!

Threads mostly on Charlemagne:

• "Pretty much anyone with Western European ancestry is descended from Charlemagne." From the TIL thread about Christopher Lee. Is this true?

• What is the chance someone of Continental European stock is related to Genghis Khan? Charlemagne? Others?

• Everyone knows that 0.5% of the population is descended from Genghis Khan. Are there statistics on other prolific leaders?

• Royal Bloodlines

and Genghis Khan:

• How realistic is the Genghis Khan genetics theory?

• 1 in every 200 men is a descendant of Ghengis Khan. Is this really true?

• How accurate is this: "8% of Asians are direct descendants from Genghis Khan?"

• Are there more direct descendants of Ghengis Khan himself than Jewish people?

• Did the Mongol Empire change the conquered populations racially?

• Women's perceptiom of Gengis Khan?

• How did we estimate that there at 16 million descendants of Genghis Khan living today?

N.B.: some caveats when reading these threads, since comments are not always clear about what they are referring to:

• "Descendants" are a given person's children's children's children's ... children. As a hypothetical example, imagine that after 2 generations he has 4 children (2 sons and 2 daughters), each of whom have 4 children (2 sons and 2 daughters): that's 4 + (4*4) = 20 descendants.

• "Male-line descendants" are a given person's sons' sons' sons' ... sons only. Using the above example, after 2 generations he has 2 + (2*2) = 6 male-line descendants (that is, his two sons and their sons; his daughters, his sons' daughters, and his daughters' children are excluded). This is the group that Y-chromosome DNA evidence refers to (see next point).

• Y-chromosome DNA evidence does not prove descent from a given person personally, but from people with genetic markers which he is assumed to have shared

edit: typos