My Jackson Line - Connection to the Mayflower and Ol' Stonewall?

Speaking of my Pennsylvania ancestors, Charles' wife, Viola Gertrude Peck was the daughter of Lura Ann Jackson.  I've discovered Lura's father was Joshua Jackson of Potter Co., Pennsylvania, and her mother was probably Elsie Rounds (Joshua's second wife).  This week another descendant of Joshua, Linda Smith, contacted me.  She believes her ancestor, Lucinda Jackson, a (half) sister of Lura, was the daughter of Joshua's first wife, Miriam Standish...yes, THOSE Standishes, of Mayflower fame!  Linda quoted a biography of one of Lucinda's grandchildren that states the Standish connection, as well as making the claim that Joshua was related to Stonewall Jackson.