I did CompSci and virtually everyone either had a high powered laptop, desktop, or both.
That said usually CompSci departments have very powerful desktops available to students so for work purposes you wont really need to.
But for fun? Yeah definitely bring it.
As for gaming clubs: Didn't go to Essex, but I've never heard of a Uni that didn't have one. Don't worry, plenty of other geeks will be there, with plenty of clubs for you.