Zinc evaporation/toxicity at room temperature
Zinc will amalgamate with mercury, but whether that reduces the vapor pressure of mercury very much is debatable. At best, the increased volume of zinc amalgam and the ability to amalgamate with fine droplets of mercury would make it easier to clean up the mercury as amalgam.

It might be better to spread some sulfur dust (flowers of sulfur) over the area. The sulfur should react with the mercury and tie it up as sulfide, with very low vapor pressure. Sulfur powder (and possibly the HgS) might also be easier to clean up than zinc dust, because sulfur is more like ordinary dust than a dense metal dust.

BTW, I love mercury thermometers. And I hate it when I break them!