React JS react-router-dom results in 404

I am fairly new to React and am trying to utilize the react router feature. I am displaying Static pages on a TV screen pointing to a certain URL.

When in my Dev environment everything is working fine. When I enter the correct path it points to the set component. When loading it onto my hosting server and testing on a Live URL pointing to the desired path results in a 404.

import React, { Component } from 'react';
import './style/App.css';
import Header from './js/components/header';
import Menu from './js/components/menu';
import MenuTwo from './js/components/menuTwo';
import {  BrowserRouter as Router,  Route, Switch, Link } from 'react-
router-dom';

class App extends Component {
  render() {
   return (
     <Router>
  <div>
      <Header headerProp="maxwell's pizzeria" /> 
      <Route path="/" exact component={Menu} />
      <Route path="/menuTwo" exact component={MenuTwo} />

  </div>

  </Router>
  )
 }
}

export default App;

The webpack.config which I am also not very familiar with is as follows:

var webpack = require('webpack');
var path = require('path');

var BUILD_DIR = path.resolve(__dirname, 'public');
var APP_DIR = path.resolve(__dirname, 'src/build');

// Existing Code ....
var config = {
 // Existing Code ....
 module : {
  loaders : [
   {
    test : /\.jsx?/,
    include : APP_DIR,
    loader : 'babel'
  }
  ]
 }
}

module.exports = config;