Andrew Jackson’s parents were Irish.
Thomas Jefferson’s mother was English.
James Buchanan’s father was Irish.
Chester A Arthur’s father was Irish.
Woodwrow Wilson’s mother was Enlgish.
Herbert Hoover’s mother was Canadian.
You list people who were not under the "natural born citizen" clause like Thomas Jefferson. The other's parents were naturalized before those persons were born with the (probable) exception of Chester A Arthur who lied, obstructed and later burned his personal papers.
Every President born before the adoption of the Constitution in 1787 was eligible because of the grandfather clause of Article 2, Section 1:
“No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President;”
Both Andrew Jackson and Thomas Jefferson are covered by that clause.
James Buchanan’s father emigrated to the U.S. from Ireland in 1783. When the U.S. Constitution was adopted in 1787 (and ratified in 1788), it conferred citizenship upon everyone living in the States. Therefore James Buchanan’s father was a citizen of the United States when James Buchanan was born in 1791. So, James Buchanan was a natural born citizen.
Woodrow Wilson’s mother became a U.S. citizen by Congressional act in 1855. Wilson was born in 1856. Therefore, both of Wilson’s parents were U.S. citizens, which makes Wilson a natual born citizen.
Herbert Hoover’s mother became a citizen in 1870 under the same 1855 Congressional act that conferred citizenship on Wilson’s mother. Hoover was born in 1874 to two U.S. citizen parents and he is therefore a natural born citizen.
Chester Arthur is a different story. So, I’ll refer you to this site for the details: http://naturalborncitizen.wordpress.com/2008/12/
BTW, you forgot Andrew Johnson, but he too was a natural born citizen.
Jefferson born 1743 and Jackson born in 1767 are both covered by the “grandfather clause”.
Those parents of Buchanan, Arthur, Wilson and Hoover were all known or believed to be US citizens at the time the future Presidents were born. Arthur’s father may not have been naturalized at Arthur’s birth, and there is an appearance of an attempt to cover up that fact by Arthur.
The Judge, Horace Gray, who made the questionable assertions as to NBC in Wong Kim Ark was appointed in 1881 by Arthur.