"I am building a power supply rated 20V 4A, and I need a DC OK signal.
Could they also be powered from the supply output?"
Maybe, with a proper circuit design. You would have to design it such that the "DC OK" signal is in a NOT OK state for any of your fault conditions (supply not working, over current or over voltage trip, output out of range). We usually use an opto coupler for this output, configured so that the opto is off under any fault condition.
Note on A Design Option:
Many of our systems have multiple power supplies providing a half dozen or more regulated voltages. We have one of the power supplies, usually the first one that's turned on, generate a secondary referenced voltage of 12 V to 15 V. This voltage is then distributed to the other supplies for them to use to power their secondary referenced monitoring circuits such as over/under voltage detection, over current detection etc.
This eliminates a chicken and the egg scenario which could occur if the local secondary voltages were used to power their fault detection circuits, in that is guarantees proper operation of the fault detection circuits regardless of what's happening with the local voltages.