Leslie is correct.They are not known to be of the same family and certainly Josias is not known to be a son of Francis.
Robert Charles Anderson, in his 1995 _The Great Migration Begins_cited George Ernest Bowman's comments of 1901 as still being definitive.In a footnote in volume 3 of _Mayflower Descendant_ Bowman presented a simple and convincing argument.Not only had he (and he must have been the premier student of the Mayflower in his time if not of all time) never found "a single record which even remotely indicates the existence of any kinship," but he had found several records which prove "conclusively that they were not father and son." Of these Bowman cites three. There is a court case in which Francis was the plaintiff and in which Josias sat on the jury and Bowman points out that the court would not allow a son to sit on a jury judging the father.Bowman cites an agreement which the court would have required to be signed by all of Francis's living children which Josias, who was alive, did not sign, and a deed which states that it is made by all the children and which does not include Josias.(MD 3:97)
However there are old books, some made newly available on CD and online which make the claim that Josias is a son of Francis.One such is _Signers of the Mayflower Compact_ by Annie Arnoux Haxtun, which was first published in 1897-1899 and was reprinted by Genealogical Publishing Company and made widely available to genealogy hobbyists (such as myself) on FTM CD#23, which I'd guess is probably the source used in some of the online trees you have seen.Haxtun says, "The argument now [that Josias is not the son of Francis] is founded not upon what was said, but what was not.Certainty is self-evident, requires no assertion, so it seems to me it was not necessary to mention Josias as a son."(Haxtun, p. 54)Well, there were times when it would have been necessary to mention that he was a son, and as Bowman demonstrated, he was not.
I have a feeling you knew all this, and it is good to bring to the attention of people just starting out, as we all did at some time, the need for good quality evidence.