The question of exclusivity is the start of a somewhat slippery slope. Generally it is better to investigate the possibility until it is proven, than to make the assumption and then look for contradictions. Just from a brief investigation, it appears that the number of early listings – pre 1600 – in various parts of England are too numerous to derive from a few recently immigrated refugees. While some names and some families certainly do have singular sources, it is also true that many surnames and those families have derived from different, multiple sources. It is necessary that you establish for yourself the details of your specific family ancestry, rather than just follow a particular surname.

There is a listing that has been extracted from the Church of England parish register of Pettistree in Suffolk for the baptism of Jone Jaye in 1541. It is in the LDS archive. Time-wise, relating to Jean Calvin and Huguenot origins, this is in the ballpark, but the location is hardly one where Huguenots would be found at that time. Better luck perhaps with Jehan Le Jay (1591-) in Canterbury at the Walloons or Strangers Church, also through that source. It seems unlikely to me that this evidence will support a single family source for this surname.

Regards,
KA