Reverse light switch location 2006 Century

Assuming you've already gone through your fuses and relays, did you check that switch? It's possible they simply forgot to plug it back in. Those also use Weatherpack connectors typically, which are kinda crappy and failure prone. Pull up the boot, find the Weatherpack plug, disconnect it... one of those poles should be hot. If it isn't, check your fuses and relay.
Jump a wire across them and see if your reverse lamps come on. If not, your problem is down the harness somewhere. Either way, I'd recommend you remove the switch and use a digital multimeter to ensure it has continuity when depressed.