A memorial plaque that includes the name of John Spooner lists 1628 as his death date under the quote:
During the Pilgrim Fathers' Leiden exile, more than thirty family members died. Many were buried in the Pieterskerk along with their Leiden neighbors. "But now we are all, in all places, strangers and pilgrims, travelers and sojourners..." Robert Cushman, Pilgrim Leader, 1622.
Having said that, however, in the list of the dead, immediately following John Spooner - 1628, is John & Jane Spooner's child - 1630, which begs the question whether a) John is the same John who died in 1628; b) & c) Jane and Anne might be one and the same person, [Note: as is cited on her profile, I believe, but also on John's on FamilySearch: Jane Peck and John Spooner were apparent co-workers, in a Leiden brewery, circa 1609-1620; and later betrothed, citing Pilgrim Homes in Leiden] & whether Anne might have attempted the trip to Plymouth, but only to return to Leiden (maybe because her husband died, en route if the FindAGrave plaque connotes death in travel?), and later marrying Powell...
Thus, we have some squishy, soft evidence that a John Spooner died en route to Plymouth or at the very least in the Leiden community, depending upon how the "during the Pilgrim Fathers' Leiden exile" plaque is interpreted, but nevertheless, was buried at Leiden.
The list of the dead also includes two Peck children (Anne's maiden name), who died in 1619.
Noting that this FindAGrave memorial has the same death date as that used on John Spooner's profile. Death location is not actually specified, just burial at Pieterskerk in Leiden.
The FindAGrave memorial provides a short narrative, stating that Anne Peck was his wife (and elaborating upon her relationships); and, that a 20 Feb 1627 document stated he was about 40 years of age.
Other notations on John's FamilySearch profile (unless it's conflation) suggest that he was closer to a Norfolk birth; at least that was where he first married 5 Jun 1609 Susan Bennett by whom he had one child, Francis, also in Norfolk.
Others also say that he then he left for Leiden.
A researcher at FamilySearch visited the Leyden City Archives and leaves John Spooner Notes and indicates that Susan was the 1st wife by whom Thomas was also born, and that Thomas accompanied step-brother William and their mother Anna to America in 1637...
Not much to go in in terms of solid proofs, but the circumstantial evidence for John and Anna v. Jane Peck having married (proof exists on a Leiden marriage for John and Anna, and Anna's later remarriage, after John died) and now a line on under what circumstances the various attempts by the family to migrate occurred, leaves us in at least a position of Uncertain parents for William, but I do not think they should be detached, just yet.