The following book discusses evidence for the existence of some of the children:
Spear, Burton W. (1985), Search for the Passengers of the Mary & John, 1630, Volumes 24-27, p. 7; Google Books (snippet view).
Some key extracts are below (note that Robin Bush was Deputy Archivist at the Somerset Records Office):
"Robin Bush found a record dated 10 April 1554, drawn up at the time of his being deprived of his benefice. By this William "Squyer", rector of Charlton Mackrell, was described as aged 52 (i.e. born c. 1501-2), as having married Alice Skarlett three years before (c. 1550-51) and was ordered to be deprived of his parish (D/D/Vc 66, f.10). The implication is that he continued to live with Alice (whose identity is now revealed for the first time) and produce children since, if he had agreed to separate from her, he would probably have continued as rector."
"Robin Bush also found more information on two brothers of Henry Squire, Edith's father, William & Andrew Squire. On 16 July 1611 evidence was given in a Consistory Court case by Willian Squyor of Charlton Mackrell, farmer, aged about 60 (born c. 1551), where he lived from birth (Ref: D/D/Cd 44). At another case on 23 June 1613 evidence was given by Andrew Squier of Kingweston, blacksmith, where he had lived for 5 years, before that at Fiffett (i.e. Fivehead) for 12 years, aged about 60 (born c. 1553), born at Charlton Mackrell."
So there is evidence that William and Andrew were born in Charlton Mackrell during the time that Rev. William Squire was vicar (and married to Alice Skarlett). Of course that does not mean that they are his children, or even that they are brothers. As for Henry, again the link seems circumstantial, based on his being born around 1564, after the Rev. William had been listed as rector of Charlton Mackrell in 1561.