I have figured it out. I need to create a clone of the objects, instead of just referencing them. So here is the full source code.

CONTROLLER

angular.module('umbraco').controller('CtaBoxController', function ($scope) {

    if (!$scope.control.value || $scope.control.value === '') {
        $scope.control.value = {};
        Object.assign($scope.control.value, $scope.control.editor.config.defaultValue);
    }

    $scope.openDialog = function () {
        var editModel = {};
        Object.assign(editModel, $scope.control.value.button);
        $scope.overlay = {
            title: 'Edit CTA Button',
            view: '/App_Plugins/CtaBox/dialog.html',
            editModel: editModel,
            btnColors: $scope.control.editor.config.btnColors,
            show: true,
            submit: function (model) {
                $scope.control.value.button = model.editModel;
                $scope.overlay.show = false;
                $scope.overlay = null;
            },
            close: function () {
                $scope.overlay.show = false;
                $scope.overlay = null;
            }
        }
    }
});

EDITOR

<div class="cta-box" id="{{::$id}}_cta" ng-controller="CtaBoxController">
    <div class="card">
        <div class="card-body flex-fill">
            <h4 class="card-title"><textarea id="{{::$id}}_title" name="{{::$id}}.title" type="text" class="card-title-control transparent" ng-model="control.value.title" placeholder="the title of the box" title="Edit Title"></textarea></h4>
            <p class="card-text"><textarea id="{{::$id}}_subheader" name="{{::$id}}.subheader" class="card-sub-header transparent" type="text" ng-model="control.value.subheader" placeholder="the sub header for the box" title="Edit Sub-header"></textarea></p>
            <div class="button-block" ng-style="{'text-align': control.value.button.position}">
                <div class="btn btn-rounded" ng-class="control.value.button.cssclass.value" role="button" data-href="{{control.value.button.link}}" data-target="{{control.value.button.blank}}">{{control.value.button.text}}</div>
            </div>
        </div>
    </div>
    <div class="show-url-button" ng-click="openDialog($scope.control)" title="edit button link"><i class="icon-link"></i></div>

    <!-- Edit Overlay -->
    <umb-overlay ng-if="overlay.show"
                 model="overlay"
                 view="overlay.view"
                 position="right">
    </umb-overlay>
</div>

OVERLAY (DIALOG)

    <div>

    <!-- Button Text -->
    <div class="control-group umb-control-group">
        <div class="umb-el-wrap">
            <div class="controls">
                <label class="control-label" for="text">
                    Button Text
                </label>
                <input type="text" ng-model="model.editModel.text" name="BtnText" />
            </div>
        </div>
    </div>

    <!-- Button Link -->
    <div class="control-group umb-control-group">
        <div class="umb-el-wrap">
            <div class="controls">
                <label class="control-label" for="link">
                    Button Link
                </label>
                <input type="url" ng-model="model.editModel.link" name="link" />
            </div>
        </div>
    </div>

    <!-- Button Color -->
    <div class="control-group umb-control-group">
        <div class="umb-el-wrap">
            <div class="controls">
                <label class="control-label" for="color">
                    Button Color
                </label>
                <select ng-model="model.editModel.cssclass" ng-options="option.label for option in model.btnColors track by option.value">
                </select>
            </div>
        </div>
    </div>

    <!-- Button Position -->
    <div class="control-group umb-control-group">
        <div class="umb-el-wrap">
            <div class="controls">
                <label class="control-label" for="position">
                    Button Position
                </label>
                <input type="radio" name="position" value="left" ng-model="model.editModel.position" />&nbsp;Left&nbsp;&nbsp;
                <input type="radio" name="position" value="center" ng-model="model.editModel.position" />&nbsp;Center&nbsp;&nbsp;
                <input type="radio" name="position" value="right" ng-model="model.editModel.position" />&nbsp;Right&nbsp;&nbsp;
            </div>
        </div>
    </div>

    <!-- Button Target -->
    <div class="control-group umb-control-group">
        <div class="umb-el-wrap">
            <div class="controls">
                <label class="control-label" for="target">
                    Open in new window?
                </label>
                <input type="checkbox" ng-model="model.editModel.blank" value="{{model.editModel.blank}}" name="target" />
                Yes, open in new window
            </div>
        </div>
    </div>

</div>

PACKAGE MANIFEST

   {
  "gridEditors": [
    {
      "name": "CTA Box",
      "alias": "ctaBox",
      "description": "Call-to-action box",
      "view": "/App_Plugins/CtaBox/editor.html",
      "render": "/App_Plugins/CtaBox/ctaBox.cshtml",
      "icon": "icon-shipping-box color-green",
      "config": {
        "defaultValue": {
          "title": "This is a title",
          "subheader": "Subheader here",
          "button": {
            "text": "Click me!",
            "cssclass": {
              "value": "btn-primary"
            },
            "link": "http://umbraco.com",
            "blank": false,
            "position": "center"
          }
        },
        "btnColors": [
          {
            "label": "Default",
            "value": "btn-primary"
          },
          {
            "label": "Green",
            "value": "btn-success"
          },
          {
            "label": "Orange",
            "value": "btn-warning"
          },
          {
            "label": "Gray",
            "value": "btn-info"
          },
          {
            "label": "Red",
            "value": "btn-danger"
          }
        ],
        "btnPositions": [
          {
            "label": "Left",
            "value": "left"
          },
          {
            "label": "Center",
            "value": "center"
          },
          {
            "label": "Right",
            "value": "right"
          }
        ]
      }
    }
  ],
  "javascript": [
    "/App_Plugins/CtaBox/editor.controller.js"
  ],
  "css": [
    "/App_Plugins/CtaBox/style.css"

  ]
}