The Free Dictionary says;

Shick-shack Day - The Oxford English Dictionary suggests that this day takes its name from a corruption of shitsack, a derogatory term for the Nonconformists, Protestants who did not follow the doctrines and practices of the established Church of England. It was later applied to those who did not wear the traditional sprig of oak on May 29, or Royal Oak Day — the birthday of Charles II and the day in 1660 on which he made his triumphal entry into London as king after a 12-year interregnum.

Shick-shack has since become synonymous with the oak-apple or sprig of oak itself, and May 29 is celebrated — particularly in rural areas of England — in memory of the restoration of King Charles and his preservation in the Royal Oak. Also called Oak Apple Day, Oak Ball Day, Bobby Ack Day, Yack Bob Day, Restoration Day, or Nettle Day.

Shik-shak does sound like reduplication - the repeating of parts of words to make new forms. There are various categories of this: rhyming, exact and ablaut (vowel substitution). Examples are, respectively, okey-dokey, wee-wee and zig-zag. (www.phrases.org)

Perhaps ack does represent oak. See Bobby Ack Day and Yack Bob Day above. There are several local dialect words for the oak tree; Chitjack, Cups and Saucers, and Oak-mas and plenty of folklore concerning the tree. (Legendary Dartmoor)

Oak Apple Day; Everywhere in England sprigs of oak were worn on 29 May, Oak Apple Day. They were sometimes called chitchat and in Hereford it was 'Chit-chat Day'. According to the English Dialect Dictionary both Chit Jack and Shitsack were in use at Barford, the next village to Wishford. These customs clearly link 29 May with Jack in the Green, Robin Hood, May Day and Summer Finding. (www.odinicrite.org)

So perhaps Chit-Jack and Shit-sack have become confused or combined.

The attendants at Garland Day carry bunches of oak twigs and nettles. You can buy one or receive the other free.