I'm just making wild guesses so take it for what it is.
The bike rack, if you load your bike on the rack how far the street is from the rear wheel? The reason I asked that is because less expensive racks can make a bike bounce quite a bit on roads, especially if you hit a dip, causing the rear wheel to contact the pavement briefly resulting from either a bent wheel or some sort of damage to the hub. Then going undetected you ride the bike and the hub puts stress on the chain and eventually breaks it. What surprises me though is that on the very first ride after the hub was damaged you didn't notice something odd.
The only other thought I have is if the bike shop re-lubed or replaced the bearings in the hub, they might have failed to adjust the hub properly after they were finished with it. If you took your bike in for a tune-up then that could be what happened, or the hub simply just loosened up for some reason and neither you nor the bike shop discovered it, which the bike shop should have found it when they replaced the chain, which takes me to my first paragraph as the culprit.