This share, \\COMPUTER\c$
, is one of the Administrative shares created automatically by the system. These shares cannot be accessed by users without administrative privileges, so there should not be the problem you described. (Have you actually tested this on a normal user?)
Administrative shares can't be permanently removed, but they can be disabled:
[HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\LanmanServer\Parameters]
"AutoShareWks"=dword:00000000
However, you probably shouldn't do that, because it may cause problems.
Also, your users won't have permissions to everything through any share without also having NTFS permissions. Even Domain Administrators can't access user's home folder through an administrative share because by default there is no NTFS permissions for that.
Share and NTFS Permissions on a File Server
Access to a folder on a file server can be determined through two sets
of permission entries: the share permissions set on a folder and the
NTFS permissions set on the folder (which can also be set on files).
Share permissions are often used for managing computers with FAT32
file systems, or other computers that do not use the NTFS file system.
Share permissions and NTFS permissions are independent in the sense
that neither changes the other. The final access permissions on a
shared folder are determined by taking into consideration both the
share permission and the NTFS permission entries. The more restrictive
permissions are then applied.