I agree with Dave’s comment that you may have to follow this one up with ESRI. ArcMap has very limited options for silent install of add-ins, and they both have issues that make them not straightforward.
As far as I know, there are two options:
1) Place the .esriAddIn file in the folder location that ArcMap searches when it opens. The problem with this is that the directory is “C:/Users/<username>/Documents/ArcGIS/AddIns/Desktop<version number>”, so you can’t use this method to install for all users
2) AddIns can be installed from a shared network location (this is documented in the installation section of the ArcRORB manual). Setting the network location to look in is usually done in the ArcMap interface, but it can be done by editing the registry. I can’t offer support on this approach (nor do I recommend it unless you are *very sure* you know what you are doing), but the relevant registry key is \\HKEY_LOCAL_MACHINE\SOFTWARE\\Wow6432Node\ESRI\Desktop<version number>\Settings\AddInFolders.
Sorry I don’t have a better answer, but this is a limitation with ArcMap so we can’t do much about it. Good luck!