I sincerely doubt it. The Overview lists two daughters named Elizabeth - one born in 1599 who is connected as wife to Robert Seaver (1608-1683), married in 1634 in Roxbury, Mass. This Elizabeth, if born in 1599, was having children in 1650. It's much more likely that Elizabweth (Ballard?) Seaver was born in the 1610-1616 time frame. She came to Boston area in 1633, probably alone as a young woman (she's in The Great Migration Begins book).
The second Elizabeth listed in this profile was born in 1638 it says. That's a pretty good trick for the father at age 63, who knows whichm other and how old she is.
The three sources listed are useless - family tree material created by someone and copied endlessly. I have no sources for this Henry Ballard, does anyone?
My line is through Lydia Ballard Butterfield (Joseph), daughter of William Ballard. (Mother is Grace Berwick?). Granddaughter of Henry and Elizabeth Townsend Ballard.
A note on paternal age: a man my hiking club once impregnated the girl next door. Nothing noteworthy there, except. that he was about 72 when he did so and she was about 40. I think the oldest man to father a child was in his 90's. While you won't find a woman becoming a mother at 63, it is possible for a man that age to become a father.
Based on the nine year difference between Henry and his purported father, I severed the connection, and placed an Unknown Parents cap on him. There seems to have been another Henry Ballard born 1585, who would have been more likely to have been part of the family he was claimed to be with, which could be the source of the confusion. If primary sources or reliable secondary sources do ever show up for restoring the connection, please let me know. -Ben.