Unfortunately, Publish From and Publish To fields are really tricky to marry with the events. Technically, when you set up Publish From field in Kentico there is a system scheduled task called Content publishing which runs every X minutes to check if there is anything to publish at all. However, when this task changes the state of the page, it doesn't fire any event you can hook into and the page silently appears or disappears, i.e. none of these events is fired.
As a workaround, you can build your own scheduled task which will be checking every X minutes were there any new pages published or unpublished, and if any - do your custom logic sending to the API. Unfortunately, with the workflow it wouldn't work I think.