Frank, Henry Alden of Dedham is a separate Alden Family and not related to John of the Mayflower folks.
What you were told as a child was correct then, but not now, and being a descendant of Henry of Dedham I have looked into how this confusion arose. The problem started when the John Alden Society published Henry of Dedham as a Mayflower descendant in 1800s. As so many confused Henry Alden descendants were still applying for membership, the John Alden Society a few years ago published Henry's genealogy in the Mayflower Descendants 43/44 'Henry, the Non Mayflower Alden", which disproved any family connection.
The information that Henry of Dedham was a grandson of John Alden, was published by the society in 'Memorial of the descendants of the Hon. John Alden' by Ebenezer Alden, MD.1867 - (my copy purchased c1867 by Louisa Alden) John is listed as have 3 unknown younger children, and on p4 the children of Capt. John Alden of Boston '14th child - Henry? #15 youngest child of Capt. John Alden of Boston, eldest son of John of the Mayflower.'
p8 '#15 Henry Alden of Dedham. 'Dr. Wight informs me 'that Henry Alden went to Dedham after 1700..... 'married Deborah....' plus note; 'Thayer said that he was probably son of John Alden of Boston....but neither is baptism found in Boston, nor is he or his son mentioned in the will'.(written pre. 1867)
In the Dedham Records vols. c 1880 there is mention that Henry was the nephew of Capt. John 2 and son of David 2 Alden, sons of John and Pricilla, David died intestate.
NEHGR vol. 52 Apr. 1898 p 163 'Alden Genealogy' ... '...from Mr. Porters' article. Two of his sons (Capt John) John Jr and Nathaniel, were mariners also and tradition has it that his younger brother, Zachariah and his nephew Henry, son of David, were with him often on his voyages....."
NEHGR vol. 85 1931 p 330 Memoirs.... Mrs. Joseph Baker of Fall River, dec. of John of the Mayflower, thru David and Mary (Southworth) Alden's son Henry 2 of Roxbury, Dedham and Needham..... This was how it stood until the recent MD 43/44 article, maybe being published in a Mayflower book, (not in NEHGR) makes for more confusion of the Dedham, Needham family. Also check the string on this forum #283- Oct 1999 for we probably came from Herts, not Southampton, Eng. as was John's home. Dr. Thomas Alden of NH is prob. being Henry's son.My John 3 Alden married 1754 Mary Adams, gt. granddaughter of Henry Adams of Braintree.
Hope this helps in clearing up the misunderstandings between these two separate Alden families.