After some investigation with a simple test scene, it seems CameraClearFlags.Depth does work in Flash -- but only if the viewport of the camera fills the entire screen. If the viewport fills a portion of the screen then setting depth-only clear will clear the camera to the background colour (wiping out everything rendered to that region by previously rendered cameras).

This is in Unity 3.5.2. I haven't tested other versions. I filed a bug report.

EDIT: Until this bug is fixed, here's a workaround ...

Add the following shader to your project. It will set the depth buffer without touching the colour of any pixels. Create a material using this shader and attach it to a plane behind all the geometry viewed by your overlay camera. This plane will be drawn first, filling the depth buffer and allowing the contents of the overlay camera to be drawn with a "fresh" depth buffer.

Shader "Custom/DepthClear"
{
    SubShader
    {
        Tags {"Queue"="Background-1"}

        Pass
        {
            ZWrite On
            ZTest Always
            AlphaTest Off
            Cull Off
            Blend Zero One
            ColorMask 0
        }
    } 
    FallBack "Diffuse"
}

I don't see a good reason why it shouldn't work. Are you sure you have the correct camera-depth (order) setup?

If it really doesn't work, i would call it a bug.