What you want to do is start with a point layer, and edit the point symbol. You can find some starter styles by clicking the "Style..." button. Make you edits. Then save that symbol to a style (use the burger button on the top right next to 'Format Point Symbol'.
Once saved, the symbol will be available for use as a callout point symbol. Click the dropdown with the shield in the image below, then click 'more point symbols' and you'll see your saved symbols.