She survived her husband and is named in his 1661 will.
Who was Judith, wife of John Rogers?
Theory #1: French
In the book, "The Ancestry of Eva Belle Kempton 1878-1908/Part 1 The Ancestry of Warren Francis Kempton 1817-1879" by Dean Crawford Smith, it reads:
Deacon John Rogers, linen weaver, born say 1610; died Weymouth, MA 11 Febraury 1660; married Judith (___), living 30 April 1660 when her husband's will was proved (Suffolk Probate #264). It has been suggested that Judith was the Judith French, age 20, who embarked for New England at Weymouth 20 March 1635/36 with Joseph Huss, minister of Somerset, in whose family she was a servant (Hotten's Lists, 285). If this is the case, then John Rogers must have met and married her in a very short time after his arrival in New England. No independent evidence supports this identification.
When John and Judith's eldest daughter, Mary, took administration on her late husband John Rane's estate, she was joined by her brother John Rogers, and Stephen French, of Weymouth, as surety. The involvement of Stephen French has fuelled speculation that Judith was a French.
Speculation from The Great Migration Begins and The Ancestry of Eva Belle Kempton, summarized by Janet Bjorndahl:
It has been suggested that Stephen French, Jacob French, Mary (French) Randall and possibly Judith (____) Rogers of Weymouth, Massachusetts were the children of Richard French of Misterton, Somerset, England. Records, including the wills of Stephen and Jacob, imply that at least the first three were siblings, half-siblings or otherwise closely related. Given that their names are the same as the names of four of Richard French's children baptized in Misterton, it seems a reasonable speculation that the four in Weymouth may have been the children of Richard French of Misterton.
According to The Great Migration Begins, another possible sibling is Margaret (____) Sales, wife of Edward Sales, although it seems more likely that it was Edward's third wife (name unknown) who was a sibling (given that the reference in Stephen French's will was made in 1678/9, long after Edward's first wife Margaret had been banished). If John Rogers' wife Judith was one of the siblings, it is possible that she was also Edward's third wife, since she was widowed in 1660/1 and Edward's second wife died in 1664.
However, the identification of John Rogers' wife Judith with Judith French, daughter of Richard French of Misterton is a bit problematic. Part of the basis for the speculation that Judith (____) Rogers was a French is that there was a Judith French who immigrated to New England as a servant in the family of Rev. Joseph Huss, minister of Somerset, in 1635/36. This Judith's age was given as 20 in 1635/6. Richard's daughter Judith was baptized 1605/6 and therefore would have been 10 years too old to be the Judith French who immigrated, unless there was an error in the age of the immigrant. It is, of course, possible that the Judith French who was baptized 1605/6 died and a younger sister was given the same name, but there appears to be no record of this, and a younger sister would have had to be a twin of Jacob French (baptized Aug 1615) in order to be 20 in 1635/6.
Theory #2: King
The only reference to the "King" family is with John Rogers of Marshfield. His son John Rogers Jr, married Rhoda King. In searching the records in searching "History of Weymouth, Massachusetts Weymouth, Mass.: Weymouth Historical Society, under direction of the town, 1923, 1845 pgs." in the family genealogy section under the King Family there is no mention of a Judith King. In the same publication one can find Stephen French with a sister Mary Randall and sons Stephen and Josiah. Again no mention of a Judith.
Therefore, there is no confirmation of Judith's last name.
In "The John Rogers families in Plymouth and vicinity " Page 8, It says Judith is probably John Rogers 2nd wife.
Conclusion
There is not enough evidence to give Judith either surname. Until more evidence is found Judith should be Unknown.
Sources
See also
Savage, James. Genealogical Dictionary of the First Settlers of New England
Drummond, Josiah H. The John Rogers Families in Plymouth and Vicinity. Read before Maine Historical Society, Dec. 19, 1895; 26 pp.
King-5388 and Unknown-408222 do not represent the same person because: On the grounds that there may be a Judith King, dtr of George etc even though unsourced.
French-5236 and French-442 appear to represent the same person because: It might make sense to consolidate this Unsourced profile, as it fits one of the 2 possible identities in 442