No need to break the chain, you just need to do a little disassembly.

1. Loosen all of the nuts that you can see and remove the chain tugs that pull the hub toward the back of the bike.
2. When you have done that, you'll be able to shift the hub forward a half inch or so.
3. When you have done that, you'll be able to remove the chain from the rear sprocket and slide the whole hub back and out of the bike.