You can send files, by modifying body argument of action call.
import ReactResource from 'react-resource';
// User model
const User = new ReactResource('/api/users/{:id}', { id: ':id' });
// User instance
const user = new User({id: 10});
// Make FormData
const input = document.querySelector('input[type="file"]');
const bodyData = new FormData().append('file', input.files[0]);
// Send request
user.$update({}, bodyData);