Assuming that you've filled the dropper to the same level and squeeze the dropper at the same rate, the size of the drop when it separates from the tip of the dropper and falls will be based primarily on two quantities. The first is the surface tension of the liquid and the second is the density of the liquid.
The density is the easiest part to explain. The higher the density, the more mass you have in the same sized drop. Assuming you're doing this in a gravity environment, more mass means more weight. Which means that for the same sized drop, mercury, which has a much higher density than water, will weigh significantly more, and will therefore have more force pulling it downwards. Mercury has a density of about 13.6 g/cm3, while water has a density of 1 g/cm3.
Surface tension, in this case, provides a resistance to the downward force of gravity. Think of a glass of water which has been filled to the top. Not just full, but as full as it can be without any spilling over. If you look at the top of the glass, there's actually a dome of water which is above the top of the glass. If you were to add a drop of soap to this bubble, water would spill over the top of the glass, due to the fact that soap reduces the surface tension of the water.
Back to the question at hand, water has a much lower surface tension that mercury. Water's surface tension is about 73 dynes/cm which mercury is about 465. If mercury weren't hazardous to your health, you could sit there with a bead of mercury on your desk and basically push it around, due to its high surface tension.
So as we see, mercury is at both an advantage and a disadvantage. It has higher surface tension, which allows it to bead up more which, alone, would allow the formation of a larger drop. On the other hand, it has a higher density, which, alone, would result in a smaller drop. Which of these two factors wins out, I can't say. But the drop sizes will be different, barring a coincidence of epic proportions.
On an interesting side note, the amount of mercury and water sucked up into the dropper initially would also differ, owing to the different densities. You would get less mercury in the dropper than you would water if you squeezed the bulb equally in both cases.
Answered by: Nathan Heilman, M.S., Chemical Engineer, Baton Rouge
'To myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.'