I seek an equational axiomatization of the quantifiers of predicate logic (that permits empty domains).
Start with the equational axioms for Boolean algebra. Add the following axioms, which come in dual pairs:
Renaming (for $y$ not free in $a$): \begin{align} \forall x a &= \forall y a[x / y] \\ \exists x a &= \exists y a[x / y] \end{align}
Exchange: \begin{align} \forall x \forall y a &= \forall y \forall x a \\ \exists x \exists y a &= \exists y \exists x a \end{align}
Distributivity: \begin{align} \forall x (a \land b) &= \forall x a \land \forall x b \\ \exists x (a \lor b) &= \exists x a \lor \exists x b \end{align}
Empty distributivity: \begin{align} \forall x \top &= \top \\ \exists x \bot &= \bot \end{align}
However, there are some tautologies I can't derive, such as
\begin{align} \exists x a &\leq a & \text{$x$ not free in $a$} \\ a &\leq \forall x a & \text{$x$ not free in $a$} \end{align}
\begin{align} \exists x (a \land b) &\leq \exists x a \\ \forall x a &\leq \forall x (a \lor b) \end{align}
\begin{align} \forall x a \land \exists x b &\leq \exists x (a \land b) \\ \forall x (a \lor b) &\leq \exists x a \lor \forall x b \end{align}
where $a \leq b$ means $a \land b = a$ or $a \lor b = b$. What is the simplest way to complete or modify this equational axiomatization so that all tautologies (in the language generated by $\{\land, \lor, \top, \bot, \forall, \exists\}$) are derivable?
Some of the rules of passage fail on empty domains. For example, $Qx(\beta \land \alpha) \leftrightarrow \beta \land Qx \alpha$ (for $x$ not free in $\beta$) fails on the empty domain when $Q = \forall$ and $\beta = \bot$.
I think that Valby's paper The Universal Theory of First Order Algebras and Various Reducts answers your question.
As Greg Nisbet mentions in the comments, this work is closely related to the theory of cylindric algebras, which goes back to Tarski. But Valby works in a multi-sorted setting and allows empty structures, and I find his paper much easier to understand than the literature on cylindric algebras.