The current reality is Republic is maintaining the Anywhere clients as is, however, there are no plans to continue with development. Therefore, as things stand now, there will be no new features forthcoming.
Like other Android messaging clients, you may, of course, use Anywhere to manually delete messages on your phone. If you wish for an automated solution, I’m afraid you need to look at something other than Anywhere.
There’s no way for the user to delete anything from Republic’s Anywhere servers. My understanding is Republic doesn’t keep message content beyond 30 days. You may limit the content stored on your computer to what’s stored on Republic’s servers by using “Reset local content…”