Change the VirtualHost configuration (typically found in /etc/httpd/conf.d\vhosts.conf) by adding the following Rewrite* lines:

General Apache virtual host configuration information can be found here: Configuring Apache Virtual Hosts

<VirtualHost *:8080>
  ServerName example.com
  DocumentRoot /var/www/httpd/example.com

  <Directory "/var/www/httpd/example.com">

    RewriteEngine on
    # Don't rewrite files or directories
    RewriteCond %{REQUEST_FILENAME} -f [OR]
    RewriteCond %{REQUEST_FILENAME} -d
    RewriteRule ^ - [L]
    # Rewrite everything else to index.html to allow html5 state links
    RewriteRule ^ index.html [L]

This tells Apache to serve any files that exist, but if they don't exist, just serve /index.html rather than a 404: not found.

react-router History Reference: Configuring Your Server