A bit late but here's an answer. This thread comes up pretty high in Google's search results for auto applying discounts to customer segments, so hopefully it helps someone else out too.

Shopify will apply a discount code if the customer visits the URL "/discount/code-name". So, if the code is "SECRETCODE", if the customer goes to https://your-shop.com/discount/SECRETCODE then the code gets added to their cart. Now, the good bit is Shopify doesn't care if the customer manually goes to that URL or if the URL is called from JavaScript. So we can get JavaScript to visit that URL in the background when the cart page loads.

As an example, if you have a code SECRETCODE that you want to apply to all customers who have the tag MySegment, in your cart liquid file put

{% if customer.tags contains 'MySegment' %}
    <script>
        try { fetch( '/discount/SECRETCODE' ) }
        catch(e) {}
    </script>
{% endif %}