According to research by James Hargraves, descendant of (one) John O'Neall, there were not one, but two Henry O'Neills in Virginia during the 1700s.  Here is a comparison of James's findings for the two.

Year	Henry 1 (elder)	Henry 2 (younger)
1724	Born in Virginia
1742		Born in Christiana, Delaware
c. 1745	Married Margaret Chambers
1759	On Frederick County, Virginia, Rent Rolls
1763-1787	Birth of children:  John, 1763; James F., 1765; Rachel, 1767; Eber, 1768; Jane, 1769; William Chambers, 1772; Susannah, 1775; Henry, 1777; Thomas Asa, 1778; Hugh, 1785; Margaret, 26 Sep 1787
c. 1766		Married Hannah (--?--)
1774	Stationed at Winchester, in Frderick County Militia (Loyalist)
c. 1776	Moved to Georgia
1781-83		In Loyalist Militia.  Then moved to Florida, then returned to South Carolina, but to Union District, not Newberry (where his estate had been confiscated).
1782	Death of father-in-law, William Chambers, whose will mentions his daughter Margaret and her husband Henry O'Neill
1784-1790		Birth of daughters Nancy and Mary
1788	Died 1 May, Cumberland Island, Camden County, Georgia, adjacent to Spanish Florida.
1790-1800		Birth of daughter Hannah
1800		On census in Spartanburg, South Carolina, with wife Hannah, daughters Nancy, Mary and Hannah
1810		On census in Spartanburg, South Carolina
1815		Died 25 Jan, in Spartanburg

In summary:

1.  Henry 1 married Margaret Chambers and had the children indicated; he also died in 1788 on Cumberland Island, near Spanish Florida. Hence, he satisfies the criteria (children, date and place of death) for the Henry O'Neill who founded the Florida line of O'Neills.  Also, he may well have been the Henry O'Neill who died as described by John Belton O'Neall in the general area of northeastern Florida. 

2.  Henry 2 was born later and had a completely different set of children, logically enough.

So the question is, which one was the son of Hugh and Anne Cox O'Neall?  If we attach any importance to the dates traditionally associated with Hugh (and I'm not entirely convinced that we should), then Henry 1 was born too early to have been Hugh's son, since Hugh didn't even arrive in Delaware until around 1730.  His eldest son, William, was only born in 1734, so Henry 1 would have been his eldest son and by far!  Therefore, on this criterion, the better candidate for his son is Henry 2.