Toggle computed property.

application.controller.js

import Ember from 'ember';

// Can't use session or local storage inside this Twiddle, but you get the picture.
var _sessionStorage = {},
    // Make _localStorage = { jwt: "something" } to see that the problem is not
    // about the value "false", but rather the original value returned by the getter.
    _localStorage = {};

export default Ember.Controller.extend({
  appName:'Ember Twiddle',
  
  jwt: 'some_fancy_jwt',
  
  onInit: Ember.on('init', function() { _sessionStorage.jwt = this.get('jwt'); }),
  
  rememberMe: Ember.computed({
  	get() {
      return Ember.isPresent(_localStorage.jwt);
    },
    
    set(key, value) {
      console.debug('service:session | rememberMe.set', value);
      
      if (value) {
        _localStorage.jwt = _sessionStorage.jwt;
      } else {
        delete _localStorage.jwt;
      }
      return !!value;
    }
  }).volatile(), // Loose .volatile() to make it work.
  
  localStorage: Ember.computed(function() { return _localStorage.jwt || "undefined" }).volatile()
  
});

application.template.hbs

<h1>Welcome to {{appName}}</h1>
<label>{{input type="checkbox" checked=rememberMe}} Do you want to be remembered?</label>
<p>Current localStorage.jwt: {{localStorage}}</p>
<br>
<br>
{{outlet}}
<br>
<br>

twiddle.json

{
  "version": "0.4.13",
  "dependencies": {
    "jquery": "https://cdnjs.cloudflare.com/ajax/libs/jquery/1.11.3/jquery.js",
    "ember": "https://cdnjs.cloudflare.com/ajax/libs/ember.js/2.1.0/ember.debug.js",
    "ember-data": "https://cdnjs.cloudflare.com/ajax/libs/ember-data.js/2.1.0/ember-data.js",
    "ember-template-compiler": "https://cdnjs.cloudflare.com/ajax/libs/ember.js/2.1.0/ember-template-compiler.js"
  }
}