If you are cancelling your My BJ's Perks MasterCard, then you will need to cancel the MasterCard with the bank (855-269-1622) before your membership can be downgraded.
The bank will inform BJ's when the card has been closed and the membership will be automatically downgraded based upon your original membership type. You will need to go to the club to get your new membership card once the bank confirms that your card has been closed.
If you were a Perks Elite Member, you will be downgraded to a Perks Rewards membership.
If you were a Perks Plus Member, you will be downgraded to an Inner Circle Membership.