A simple solution would be to give your user the right to run the command without entering a password and then adding it to the list of startup programs. First, run
sudo visudo and add this line to the file (change
geomint to your actual user name):
geomint ALL=NOPASSWD:/usr/bin/VBoxClient --clipboard
Then, use the tools provided by your desktop environment to add the command to your startup applications. Instructions for Unity can be found here.
Alternatively, do it manually. Create a file called
~/.config/autostart/VboxClipbvoard.desktop with the following contents:
Exec=sudo VBoxClient --clipboard
Now reboot, and it should work as expected.