As far as I know, a user id is always needed for an enom push.
If you don't want to give away your user id, how about opening a second free account, give them that id, and then push to your own account after the 7 day period has finished (or however long it is before you can do another push).
The other alternative is to request a transfer. I'm not sure if a transfer from enom to enom works. I seem to remember this working for me once, although it might have been from one enom reseller to another one.