Yes you can flush the toilet when the power goes out by using a bucket of water.
When the power goes out, you can simply take a bucket filled with water and flush the toilet if you have no running water due to a power outage, like on a well.
Although if you're on city or municipal water supply, then you should still have running water during the power outage, assuming that the water tower remains filled or they continue to have water pressure.
But on a well, you need electricity to power the pump and so it's good to have buckets of water gathered up and even fill jugs of water up to drink and even fill your bathtub up.
When your pour a bucket of water down the toilet bowl, it does the same thing as the toilet filling up with water when you flush the toilet and the tank empties.