We investigate subsets of critical sets of some Youden squares in the context of secret sharing schemes. A subset C of a Youden square is called a critical set, if C can be uniquely completed to a Youden square but any subset of C cannot does not have a unique completion to a Youden square. That part of a Youden square Y which is inaccessible to subsets of a critical set C of Y, called the strongbox of C, may be thought to contain secret information. We study the size of the secret. Seberry and Street have shown how strongboxes may be used in hierarchical and compartmentalized secret sharing schemes.