You need to add any icons you wish to use, to a "library" for easy reference.

import React from 'react';
import { render } from 'react-dom';
import Hello from './Hello';
import fontawesome from '@fortawesome/fontawesome'
import FontAwesomeIcon from '@fortawesome/react-fontawesome';
import { faCheckSquare, faCoffee } from '@fortawesome/fontawesome-free-solid'

const styles = {
  fontFamily: 'sans-serif',
  textAlign: 'center',
};

fontawesome.library.add(faCheckSquare, faCoffee);

const App = () => (
  <div style={styles}>
    <FontAwesomeIcon icon="check-square" /><br /><br />
    <FontAwesomeIcon icon="coffee" />
  </div>
);

render(<App />, document.getElementById('root'));

Check out working code here: https://codesandbox.io/s/8y251kv448

Also, read this: https://www.npmjs.com/package/@fortawesome/react-fontawesome#build-a-library-to-reference-icons-throughout-your-app-more-conveniently

Just in case there are other idiots out there like me, make sure that you use the right name in referencing the icons.

I had

import { FontAwesomeIcon } from "@fortawesome/react-fontawesome";
import { library } from "@fortawesome/fontawesome-svg-core";
import { faUser } from "@fortawesome/free-solid-svg-icons";

library.add(faUser);

and

render() {
  return (        
    <FontAwesomeIcon icon="faUser" />
  );
}

when, in fact, the actual icon name is just "user", not "faUser":

render() {
  return (        
    <FontAwesomeIcon icon="user" />
  );
}

You can use FontAwesome icons without library like this:

import { FontAwesomeIcon } from '@fortawesome/react-fontawesome'
import { faCoffee } from '@fortawesome/free-solid-svg-icons'

<FontAwesomeIcon icon={faCoffee} />

I've installed all the necessary packages as react-fontawesome says:

$ npm i --save @fortawesome/fontawesome-svg-core
$ npm i --save @fortawesome/free-solid-svg-icons
$ npm i --save @fortawesome/react-fontawesome