For completely submerged bodies the buoyance force, being simply equal to the weight of the displaced fluid, is stronger for a denser fluid.
But you know that the buoyancy force for a partially submerged body (like a sailing boat) must be equal to the weight of the body (unless the boat sinks or starts flying like a balloon).
Since the buoyant force is equal to the weight of the displaced fluid, a (non-sinking) boat displaces always the same mass, no matter which fluid, but more volume of a less dense fluid.
A classical example happens if you submerge an egg in water. It sinks to the bottom of the top. Then start adding salt, until eventually the egg will raise. See for example Tommy's webpage:
A quite different question is if a boat would happily float in a denser fluid like mercury, without turning upside down. The shape of the submerged part is very important for the stability. The buoyancy centre must be higher than the centre of mass, otherwise it will be unstable (that is why ballast is needed in many cases, to make a boat heavier in its underwater part... too much of the boat above water would result in a dangerous high centre of mass)
EDIT: Ok, when the partially submerged body is in equilibrium, then
$$W_{\text{displaced fluid}}=W_{\text{object}}$$
$$\rho g \Delta V = W_{object}$$
Since $g$ and the weight of the object $W_{\text{object}}$ are fixed, an increase in density means a decrease in the submerged volume, for the equation to hold.
The argument sounds perfectly reasonable.
Consider arbitrary parcel of fluid in equilibrium. It exerts downward force equal to it's weight on the surrounding fluid, and it does not move. Therefore according to the second law of motion, the downward force must be balanced by upward force of equal magnitude, the buoyant force (otherwise it would start to move, contradicting the equilibrium assumption).
The buoyant force is exerted by the fluid surrounding the parcel. Therefore if we replace the parcel with something else, there is no reason for that force to change.
Best Answer
Expanding on @SebastianRiese's comment, the buoyant force already takes care of the downward force caused by the upper liquids. Lets consider the problem from a physical perspective. There is the downward force from gravitation, the downward force from the liquid above, and the upward force from the liquid above.
In the diagram, the gravitational force isn't shown but the pressures causing the upward and downward forces from the liquid are shown. The buoyant force is calculated from subtracting these two pressures and multiplying by the cross-sectional area:
Hence, the net force is
Hence, the buoyant force already takes into account the downward force caused by the liquids above.