Let $M$ be the ordered Mostowski model (T. Jech, The Axiom of Choice, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subset of $A$; then $S$ is a union of finitely many intervals. So if $S$ has a lower bound then $S$ is disjoint from (many) sets of the form $D(a)$, and if it does not have a lower bound then it contains (many) sets of the form $D(a)$. So in this model we have a linearly ordered set, $A$, such that $H_A$ does not have property $B$.

The Jech-Sochor embedding theorem (Theorem 6.1 in Jech's book) can be used to transfer this to a model of $\mathsf{ZF}$.

Let $M$ be the ordered Mostowski model (T. Jech, The Axiom of Choice, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subset of $A$; then $S$ is a union of finitely many intervals. So if $S$ has a lower bound then $S$ is disjoint from (many) sets of the form $d(a)$, and if it does not have a lower bound then it contains (many) sets of the form $d(a)$. So in this model we have a linearly ordered set, $A$, such that $H_A$ does not have property $B$.

The Jech-Sochor embedding theorem (Theorem 6.1 in Jech's book) can be used to transfer this to a model of $\mathsf{ZF}$.