Solved this problem. ATTN Rudal and all others, here is a step-by-step of how I did it.
1) Go to the folder you've installed Origin Games in (by default, C:\\Program Files (x86)\\Origin Games
2) Inside of the Mass Effect 3 Demo folder, go to __Installer (note underscore) then the DirectX folder.
3) Click on the redist folder and send it to the Recycle Bin with the Delete key.
4) Go into Origin, try to install again, and then when given the option to repair the game, do so. It will redownload all those files that just got deleted, and for me, this time they came properly signed.
5) The game should install properly.
That should do it! Happy hunting.