ESLint with React gives `no-unused-vars` errors

I've setup eslint & eslint-plugin-react. 

When I run ESLint, the linter returns no-unused-vars errors for each React component.

I'm assuming it's not recognizing that I'm using JSX or React syntax. Any ideas?

Example: 

app.js

import React, { Component } from 'react';
import Header from './header.js';

export default class App extends Component {
  render() {
    return (
      <div>
        <Header />
        {this.props.children}
      </div>
    );
  }
}

Linter Errors:

/my_project/src/components/app.js
  1:8  error  'React' is defined but never used   no-unused-vars
  2:8  error  'Header' is defined but never used  no-unused-vars

Here is my .eslintrc.json file:

{
    "env": {
        "browser": true,
        "es6": true
    },
    "extends": "eslint:recommended",
    "parserOptions": {
        "ecmaFeatures": {
            "experimentalObjectRestSpread": true,
            "jsx": true
        },
        "sourceType": "module"
    },
    "plugins": [
        "react"
    ],
    "rules": {
        "react/jsx-filename-extension": [1, { "extensions": [".js", ".jsx"] }],
        "indent": [
            "error",
            2
        ],
        "linebreak-style": [
            "error",
            "unix"
        ],
        "quotes": [
            "error",
            "single"
        ],
        "semi": [
            "error",
            "always"
        ]
    }
}

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