What I saw in the photo above seemed to be a regular nut encircling a threaded shaft; you can see a “rim.” I found a few of the nuts as you described them, and upon examining the photo again, it seems that the circular shaft end is simply wear from the cable connector; faked out.
Here are a couple of example nuts from an ebay seller; I could not find anything specific for any JangleBox:
https://www.ebay.com/itm/204052532920?c ... l4QAvD_BwE
https://www.ebay.com/itm/314088733173?_ ... 1506.m1851
https://reverb.com/item/60452984-catali ... cts-pedals
Each of these web pages advertises other similar replacement nuts. I guess you removed the output nut to get a good overview of it…. There must be a JB Nano replacement nut somewhere.
This seller has a lot; there must be one that can serve as a replacement for the JB nut; maybe contact the seller…
https://reverb.com/marketplace?query=nu ... rts&page=1
Man, this looks close. Contact the seller and ask if it would fit a JB. If not, he’ll probably mention one that will:
https://www.ebay.com/itm/312038595911?c ... Ysk0Fg72Mc