I haven't evaluated the other answers, but for my particular case, an extremely solution worked. I don't believe this is well documented, so it might be fairly common for people to still encounter this issue. For my particulars, the relevant problem and solution were identified here.
To address, add <base href='./' />
to the index.html (or whatever your starting html file is that hosts your SPA). This is a complete example of mine:
index.html
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<base href="./" />
<link rel="icon" href="%PUBLIC_URL%/favicon.ico" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="theme-color" content="#000000" />
<meta
name="description"
content="Web site created using create-react-app"
/>
<meta
http-equiv="Content-Security-Policy"
content="script-src 'Self' 'unsafe-inline';"
/>
<link rel="apple-touch-icon" href="%PUBLIC_URL%/logo192.png" />
<!--
manifest.json provides metadata used when your web app is installed on a
user's mobile device or desktop. See https://developers.google.com/web/fundamentals/web-app-manifest/
-->
<link rel="manifest" href="%PUBLIC_URL%/manifest.json" />
<!--
Notice the use of %PUBLIC_URL% in the tags above.
It will be replaced with the URL of the `public` folder during the build.
Only files inside the `public` folder can be referenced from the HTML.
Unlike "/favicon.ico" or "favicon.ico", "%PUBLIC_URL%/favicon.ico" will
work correctly both with client-side routing and a non-root public URL.
Learn how to configure a non-root public URL by running `npm run build`.
-->
<title>React App</title>
</head>
<body>
<noscript>You need to enable JavaScript to run this app.</noscript>
<div id="root"></div>
</body>
</html>