There really isn't much to know about him because he wrote very little. A couple of handbills which were reprinted in the Boston Gazette, a newspaper ad directed towards him (all in the spring of 1774), and then in 1777 an announcement that he'd returned (but then nothing else).

Thanks to the Massachusetts Historical Society the originals of the Boston Gazette from 1774 can be viewed online, so I've gone ahead and transcribed them here.

The first appearance of Joyce, Jr. was in the January 17, 1774 Boston Gazette, which is reporting on a broadside that was put up around town. The paper proceeds to quote the broadside:

Brethren and Fellow Citizens!

You may depend that those odious Miscreants and detestable Tools to Ministry and Governor, the Tea Consignees, (those Traitors to their Country, Butchers, who have done and are doing every Thing to Murder and destroy all that shall stand in the Way of their private Interest,) are determined to come and reside again in the Town of Boston.

I Therefore give you this early Notice, that you may hold yourselves in Readiness, on the shortest Notice, to give them such a Reception as scuh vile Ingrates deserve.

Joyce, jun

Chairman of the Committee for Tarring & Feathering.

If any Person should be so hardy as to Tear this down, they may expect my severest Resentment. J.Jun

Joyce Junior appeared again on January 31, 1774. Another handbill had been posted "the previous Morning", and the contents were, as usual, reprinted in the Boston Gazette

Brethren, and Fellow-Citizens!

This is to Certify, That the modern PUnishment lately inflicted on the ignoble John Malcolm was not done by our Order---we reserve that Method for Bringing Villains of greater Consequence to a Sense of Guilt and Infamy.

Joyce jun^r

(Chairman of the Committee for Taring [sic] and Feathering.

If any Peson should be so hardy as to tear this down, they may expect my severest Resentment. J.jun.

John Malcolm has the unfortunate distinction of being perhaps the most brutal tarring and feathering ever done in colonial Boston. The normal procedure for conducting a tarring was to strip the victim down to their shirtsleeves, pour the tar on them, dump the feathers over them, and then send them on their way.

Sometimes the victim was paraded through town or other common areas, but the intent was to humiliate and shame, and above all to bring them back into expected conformity with the public will. Not to cause serious injury.

On January 25, 1774 Malcolm was seen threatening to strike a boy with a cane. He was prevented from doing so by George Hewes (who had participated in the Tea Party). A verbal confrontation followed, with Malcolm telling Hewes not to get involved with the business of a gentleman, and then Hewes making a comment about Malcolm having previously been tarred & feathered.

Malcolm then struck Hewes on the head with his cane injuring him fairly severely. That night a mob formed and rug Malcolm from his home, They paraded him around all night, at points threatening to hang him or chop his ears off. The ordeal for Malcolm included beatings as well. This behavior was shocking in both the brutality of it, and because Malcolm had been drug out of his home.

Joyce's next appearance in the record isn't because of his own words but others. On March 28th, 1774 a letter was written to the publishers of the Boston Gazette.

Messieurs Edes and Gill,

Time eradicates the Horrors of Guilt, and effaces the Apprehensions of Danger: In Confidence of the Truth of these Maxims, it is said a Number of the most eminenet of the Plymouth Protestors [sic] are preparing to make a Visit to the Metropolis.

The following is a List of those who have not recanted:

Underneath this letter an advertisement appears:

A CARD

The Plymouth Protestors present their Compliments to Joyce, jun. and ask the Favor of him to make Preparation for a Reception of a select Committee from their Body, who propose to honor the Metropolis with a Visit very soon.

There's another instance where "Joyce, Jr" shows up in a 1777 newspaper, but it's merely to announce his return and then he's gone again.

As for the identification of "Joyce Jr", there's strong speculation that he's the pseudonym of John Winthrop, who had served as president of Harvard and was a noted scientist. However this is based on a document written by a Loyalist which was donated to the Massachusetts Historical Society. The document consists of a list of names of various Boston leaders who had been named to enforce the Continental Congress' Association policy. This Loyalist had made some rather caustic remarks to each of the names on the list. The original transcriber of the document reports Winthrop's name as "illegible", but later transcriptions clarified that the Loyalist was referring to Winthrop.

Sources:

The Shoemaker and the Tea Party: Memory and the American Revolution by Alfred Young. Good information on the Tea Party, George Hewes, and John Malcolm

Tar, Feathers, and the Enemies of American Liberties, 1768-1776 by Benjamin Irvin in The New England Quarterly

Proceedings of the Massachusetts Historical Society vol 12 (for identification of Joyce Jun)

Joyce Junior Once More by Albert Matthews. This paper connects the colonial Boston Joyce Junior to the figure of Cornet Joyce from the English Civil Wars.