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Be careful not to drop the eggs.
Literal
Egg [object-を] drop-not-so-as-to attention-attach-do.
~ないように ('so as not to X') + 気をつける ('be careful / pay attention') is a fixed pattern for 'be careful not to do X.' 気をつけなさい uses the ~なさい imperative — firm but not harsh, typical of adults addressing children or issuing gentle instructions. 気をつける literally 'attach one's spirit,' i.e. pay attention.