It's so long since I posted my original request that I'm not sure whether anyone will be interested in the final outcome, but in case anyone is, I thought I would post my eventual findings.
Eyedoc's helpful reply concerning the apprenticeship of John Eggett, son of Benjamin was a huge clue!
Ancestry has recently included some records from Gorleston among their "Norfolk, England, Church of England Baptism, Marriages, and Burials, 1535-1812". From these I was able to find that John Eggett was baptised 7th Mar 1735 • Gorleston with Southdown, Suffolk, England. He was the son of John (as we suspected) and Elizabeth Eggett. I haven't found their marriage record or Elizabeth's maiden name. They had one other child that I have found, Margaret, baptised 1733.
John died in 1744 and Elizabeth (Elizabeth Eggett, widow.) in 1750.
John senior-he of the very helpful apprenticeship record- was baptised 3th Oct 1703 • Gorleston-on-Sea, Suffolk, England.
He was the son of Benjamin Eggett and Margaret Stone.
Benjamin and Margaret married in Gorleston in August, 1699. They had 6 children in 21 years:-Richard, John, Margaret, Benjamin who died in infancy, another Benjamin, and finally, Rachel.
I don't have any birth records for Benjamin or Margaret, but Margaret died in 1727, described as "Margaret Eggett, wife of Benjamin" and Benjamin died in 1728.
9 years of searching and I finally found him! Many thanks to everyone who made helpful suggestions.