I'm too young to remember it but as far as I know it is very much an Everton song... we just don't sing the whole version anymore. Full lyrics are here:
https://www.evertonarentwe.com/songs-chants/we-shall-not-be-moved/ along with an alternative version about the 1966 cup final.
Onwards Evertonians, onwards for to see
See those royal twin towers our team at Wembley
See those royal blue jerseys as graceful as can be
Forging onto victory and fighting constantly
Onwards Evertonians don’t let your pride be moved
Remember our song for Wembley is we shall not be moved
We shall not, we shall not be moved
We shall not, we shall not be moved
Just like the team that’s gonna win the FA Cup
We shall not be moved
@magicjuan is this the version you meant?