I'm glad to hear you are using Asciidoctor for writing your API docs.
Assuming you are generating a standalone HTML document, the JavaScript and CSS for highlight.js is linked from a CDN by default. The built-in HTML converter does not currently support embedding these assets directly into the HTML. If you want that behavior, you can implement a custom converter or a custom document HTML template.
Of course, that's assuming you are generating a standalone HTML document. If you are generating embeddable HTML, then Asciidoctor only outputs the contents of the body element. It's up to you or the tool calling Asciidoctor to put that into an HTML template that has the highlight.js assets.
Cheers,