Twenty-six-year-old Bush joined the Clarets a year ago from Lincoln City and played an integral part of the side that reached the National League South play-off final last term.
The strong central defender appeared almost fifty times for Chelmsford last season, finding the net on the opening day of the season draw at Truro City last year.
Bush started his career at Brentford, and has also spent time with AFC Wimbledon, Gateshead and Hereford.
He has appeared in each of the Clarets’ matches this season and, with the defender under contract at the Melbourne Community Stadium, compensation has been agreed between the two clubs for his immediate release.