To remove spawn points in JJS, open Build Mode , create or select a spawn point block, then use the build menu option for Remove Spawns or Remove all spawns. A forum post and build tutorial both describe that this clears the natural/spawn points, and one tutorial notes you may need to do it before reassigning team spawns.

Quick steps

  1. Enter Build Mode in your private server.
  1. Open the build menu and find the spawn point controls.
  1. Click Remove Spawns or Remove all spawns to clear them.
  1. If you are setting up team spawns, assign the team color and turn off neutral on the spawn point so only that team uses it.

Small detail

One tutorial specifically says to remove all spawns first before placing team-based spawn points, which is why the option may seem required even when you only want to redesign the setup.

TL;DR

Use Build Mode, then choose Remove Spawns / Remove all spawns in the spawn settings. After that, place new spawn points if needed.