To get the control you want, setup a webpage on the server that would allow the upload but not the download of files.
On the file share, you could set the permission for world writeable, but not readable. Note: that you might not be able to have subfolders.
At this point the user could write into the share, but not read from it. The problem you may face is that the file share is a samba linux based and I am not sure whether the granular settings/control are easy to setup.