The question on a characterization of algebras possessing a finite basis for their equations or quasi-equations has been studied by various authors. The present paper deals with finite \(\{0,1\}\)-valued unary algebras \(\mathbf M\) with \(\{0\}\) being a subalgebra. Then each nontrivial clone operation corresponds to a basic operation of \(\mathbf M\). Now clone operations are represented by a table where rows are ordered in some natural way. The main result of the paper is as follows: \(\mathbf M\) has a finite basis of quasi-equations if and only if the rows form an order ideal. The notion of a quasicritical algebra appears as a considerable useful tool for the proof.