I don't have absolute proof that Anson was the son of Phineas Squire and Catherine/Katherine Bates, but I do have a note that Phineas and Anson are in the same plot in New Durham Cemetery aka Durham Cemetery #2 (no source written down):
Squire, Anson d. 7 May 1867 age 82
Sally, wife of Anson d. 23 Aug 1852 age 68
Squire, Chauncey R. d. 24 Aug 1843 age 18
Squire, Phineas d. 5 Dec 1839 R.W. Vet Sgt. 6 Rgt 1777-1781
It was always accepted in my family (Orin D. Squire's line) that Anson was their son.Anson married 1 Sept 1805 in North Marlborough, Massachusetts Sarah Canfield, apparently dau of Gideon Canfield and Anna Robinson.
Note that Phineas's daughter Nancy, wife of Moses Riley, had a son named Anson Bates Riley.
The places to double check this info would be the Connecticut State Library and the Connecticut Historical Society, both in Hartford.They have wonderful records.