Business hours drive three things in Message: the widget online indicator, the auto-reply for after-hours messages, and the routing rules that send after-hours phone calls to voicemail or to an on-call rotation.
Configure them
Open Settings > Business hours. Pick your timezone. Toggle each day on or off. Drag the start and end handles for each day. Save.
Multiple time zones
If your team works in multiple time zones, create one rule per department instead. Each department has its own business hours that override the workspace default.