Unfortunately, I don't have access to the production F5's, and my lab F5-VE is not configured at the moment. Though, I was able to replicate your issue another way. My Horizon environment is internet accessible and we have NAT through the front-end firewall. I can load up Horizon home page no problem using the external address. The second I try to load /admin, it gives me the same 404 you're seeing. This is with the NAT IP and with the external address.
Wild Speculation Time: The only thing I can assume is that the web consoles in Horizon are configured to answer to their specific FQDN for virtual directories. When using your VIP, you are technically accessing the Horizon server (which is why the Horizon webserver is responding with the 404) but you are not trying to access that specific named FQDN virtual directory (connection.server.com/admin). That may be why the server is responding 404, as it doesn't have a virtual directory listing for VIP.com/admin.
I hope that made sense. I personally just stick to accessing a connection server directly. Maybe someone else has a more eloquent and official answer to this.