I created a character in CC 3.4 (with Hair Builder extension), and it looks like this when I import it into iClone 7 :




Then, when the character is in iClone 7, I export it as FBX with '3DS MAX' selected in the "Target Tool Preset" dropdown box, and also ticked the "Embed Textures" box.

Then, in 3DS Max, when I import the FBX file, it looks like this:




What's the problem here?

I think it has to do with materials not being set to Alphas or Transparent. Check that. 

Thanks! Is this something that should be done in iClone or 3Ds Max?

If you save all the hair textures of that hair style, it will include opacity for the hair. You can save the eyelashes too, I think.
You are missing the opacity for hair. That is your problem. Just apply the opacity hair texture in the third party software.
Look at photos, the first one shows hair opacity strength is 0. The second photo shows hair opacity strength at 100.