To implement the above you need some kind of persistent storage of the state of that button.
It could be saved to disk so that the game can be resumed, or it could just be a local variable stored in something like a GameManager (see millions of Youtube tutorials for the idea) that sticks around for the length of a given game, as opposed to just a single scene.
When a scene loads, there would be a script on that button that would check this persistent data to decide if the button should be present or not.
Load/Save steps:
An excellent discussion of loading/saving in Unity3D by Xarbrough:
Loading/Saving ScriptableObjects by a proxy identifier such as name:
When loading, you can never re-create a MonoBehaviour or ScriptableObject instance directly from JSON. The reason is they are hybrid C# and native engine objects, and when the JSON package calls new to make one, it cannot make the native engine portion of the object.
Instead you must first create the MonoBehaviour using AddComponent() on a GameObject instance, or use ScriptableObject.CreateInstance() to make your SO, then use the appropriate JSON “populate object” call to fill in its public fields.
If you want to use PlayerPrefs to save your game, it’s always better to use a JSON-based wrapper such as this one I forked from a fellow named Brett M Johnson on github:
Do not use the binary formatter/serializer: it is insecure, it cannot be made secure, and it makes debugging very difficult, plus it actually will NOT prevent people from modifying your save data on their computers.