Joseph Jencks was one a well respected family of early Providence.

The evidence for the involvement of a Joseph Jencks comes solely (at this time) from  State of Rhode Island and Providence Plantations at the End of the Century, A History, Edward Field ed, (Boston, Mason Publishing Co., 1902 )Vol. I. p. 467.  The name of Joseph Jencks is found in his list of names and also in the list by Catherine Williams, Biography of Revolutionary Heroes: Containing the Life of Brigadier Gen. William Barton and also of Captain Stephen Olney. Providence, Self-published. 1839) p21.   Field says:

The names of only a few of these who took part in the night's work have been handed down. In the different accounts which have from time to time appeared the following names have been ascertained: Capt. Samuel Dunn, Capt. Benjamin Page, Capt. Turpin Smith, Capt. John B. Hopkins, Joseph Bucklin, Captain Shepard, John Brown, Abraham Whipple, Ephraim Bowen, John Mawney, Captain Harris, Joseph Jencks, Justin Jacobs, Simeon H. Olney, Joseph Tillinghast.

Field was a good researcher but like Williams, he gives no source for his naming of Jencks.  

Now what do we know about this "Joseph Jencks?"  At this time, nothing.  Our first problem is that in Jencks research there are name variations to be considered of Jencks, Jenckes, Jenks, and Jenkes.  All of them have been use to refer to the same families, at the same time, and even among the siblings of the same family.  It simply depended on the spelling preferences of the speaker or writer.  Joseph Jencks of 1661-1740, was a Colonial Governor of Rhode Island.  His tombstone has it as Jencks.  Both we and the  Gaspee Virtual Archives of distinguished researcher John Concannon have  standardized on Jencks as our spelling.  However, since there are the spelling variations, we have to use all of them in searching the data and possible source

John Connanon, at his Gaspee Virtual Archives, summarizes the history and genealogical leads for, but lack of definitive  evidence of, the identity of this Jencks, and we have nothing of substance to add, except for the following.

There were at least two families of Jencks in Providence in 1770, and probably thus in 1772.  One, the family of Judge William Jencks. Judge Jencks lived on the south of the Parade, very near to the 1770 location of the home of Joseph Bucklin 4th.  Although Judge Jencks died in 1765, his family continued to live there.  Judge Jencks had a son Joseph (born about 1714 in  Pawtucket) and a daughter Susanna who married Joseph Bucklin 4th, who was the father of Joseph Bucklin 5th, who that fired the shot wounding the commander of the HMS Gaspee in 1772.

The Joseph Jencks born 1714 would have been about 58 years old at the time of the Gaspee raid in 1772.  This is probably too old a man to have been in the attack,  but it would not be surprising if he had a son also named Joseph Jencks in the appropriate age range for a Gaspee attacker.

The other is the family of a John Jencks, of whom we know something about because of the materials on Dr. Ephraim Bowen. Dr. Ephraim Bowen came to Providence, from Rehoboth, in 1739.  [Carpenter 1771] (Thus, he and Joseph Bucklin 4th  probably knew each other from Rehoboth.) Dr. Bowen lived at what is now the intersection of College and Water (then Main Street).  As opportunity offered, he extended his lot up the hill away from the water.   There was an agreement between Bowen and John Jenckes, who adjoined on the north, so that one would not extend their property further  than the other. [Carpenter 1771].

John Jencks, together with Stephen Hopkins, Thomas Greene, and Benjamin Man, were the Deputies elected  in May of 1772 by Providence to serve in the state legislature.