What if Chaff were a cloud of millions of small mirrors? Drop chaff behind you and all of a sudden there's this cloud of mirrors that are reflecting laser fire in any old direction

This idea also comes up under every news about today's military grade lasers as well and
won't work. Your tiny particles would simply vaporize instantly by the massive amount of energy.
If you coat an object under a mirror surface a high energy laser will simply vaporize the coating. It is possible to manufacture materials that will adsorb and deflect the energy to some amount but they're very expensive because they're often made of rare materials.
See this section of the article.
So if you produce such a material you'll coat only a high value target with that because of the limited amount of material you can produce. And it still wont eliminate the laser, just reduce it's effect.
Assuming in the ED universe enough material would be available to produce such a coating, I think in space a small floating particle, even a small object of the size of lets say a beer bottle wouldn't stop the laser. First you have a huge space to cover, you would have to create an incredible huge and dense cloud to make it useful. Remember, you're flying hundreds of meters per second. Next issue is that a high energy laser would simply push the objects away, even if they could withstand the energy without getting vaporized.
Yes, lasers can push things.