It sounds like a Boxwood (Buxus sempervirens). Unfortunately, my limited experience propagating Boxwoods is confined to the Japanese Boxwood (B. microphylla), and to smaller cuttings--usually 3 joint semihardwood cuttings in early spring.

You've got nothing to lose in giving it a try now. I wouldn't take especially large cuttings though - just normal size and let them grow once they've developed roots (if they take, that is)

If you were to take off a good big branch and stick it in the ground or a pot, there would just be lots of leaves to support and no roots to do the job, so it's not likely to be successful.

Pull off some pieces about 4" - 6" long and put them round the sides of a pot of compost. Put in a sheltered shady place out of doors and hope for the best.