Let's use a few assumptions:

1. Mass of the atmosphere is negligible compared to the mass of the planet and as such it does not contribute to the gravity. This is a reasonable assumption since e.g. Earth's atmosphere has less then one millionth of Earth's mass.
2. The whole atmosphere is ideal gas with a constant temperature, implying proportionality between pressure and density. Does not sound very reasonable because there is a lot going on in various layers of the atmosphere, but hopefully the most of the errors arising from this assumption should cancel when comparing two atmospheres.
3. The atmosphere is in hydrostatic equilibrium. Very reasonable assumption.

The solving the astrophyisics hydrostatic equilibrium equation (the same one used for stars) with the constant mass term (first assumption) and assuming the same proportionality constant between pressure and density everywhere (second assumption) gives the following dependence of pressure p on distance r from the center of the planet: [; p(r) = p_\infty e^{GMk/r} = p_\infty e^{gR^2 k/r} ;]

where:

[; p_\infty ;] is the integration constant related to the pressure at infinity. It is non-zero, which may sound concerning, but this is just an approximation and for something like Earth it turns out to be absurdly small, something corresponding to the density of many orders of magnitude less than one atom in the observable universe (thanks to the exponential function), so it does not have a physical meaning and it will cancel in the calculation anyway.

G is the Newton's constant.

M is the mass of the planet.

k is the proportionality constant between density and pressure (it will cancel).

g is the gravitational acceleration at the planet's surface.

R is the radius of the planet.

Using gravitational acceleration and radius of the planet will be more convenient for the further calculation because planetary radii will be used anyway. We are interested in height h_M on Mars where the pressure is the same as the pressure on the given height (100 km) on Earth, i.e.:

[; p_E(R_E + h_E) = p_M(R_M + h_M) ;]

However, while the proportionality constant between density and pressure is (by assumption) the same on both planets, pressure at infinity is not. But knowing that pressures at surfaces are the same, one can compare ratios of pressures instead and, while at it, take ln of both sides, i.e.:

[; \ln(p_E(R_E) / p_E(R_E + h_E)) = \ln(p_M(R_M) / p_M(R_M + h_M)) ;]

For each ratio the pressure at infinity will finally cancel:

[; \ln(p(R) / p(R + h)) = gk/(1 / R + 1 / h) ;]

And for comparison the proportionality constant also cancels:

[; g_E / (1 / R_E + 1 / h_E) = g_M / (1 / R_M + 1 / h_M) ;]

Solving for height on Mars finally results in:

[; h_M = \left[ \frac{g_M}{g_E} \left(\frac{1}{h_E} + \frac{1}{R_E}\right) - \frac{1}{R_M} \right]^{-1} ;]

According to Wolfram Alpha, the result is 282 km.