The main issue is that the Slingbox (as any other network device) will need to get an IP address when accessing to the Internet. Since the router is the only device that provides IP address, the Slingboxes will not work with your current network layout. They all must be connected to the home router in order to get the IP address.
I have the new Slingplayer too. My public ip address has recently changed (by the cable co.) and my computer no longer can access the Slingbox. No problem with the iPad and iPhone, just the Mac Computer. Anyway, I checked my on-line settings (after going through the "setup procedure" again), and the on-line settings still show the old public address.
How do I change my on-line public address?