Should binding constants be unitless when deriving fractional occupancy equations from reactions?
The equilibrium constant does not need to be unitless, because it is depending on its definition. See goldbook:


  Equilibrium Constant
  Quantity characterizing the equilibrium of a chemical reaction and defined by an expression of the type $$K_x = \Pi_B x_B^{\nu_B},$$ where $\nu_B$ is the stoichiometric number of a reactant (negative) or product (positive) for the reaction and $x$ stands for a quantity which can be the equilibrium value either of pressure, fugacity, amount concentration, amount fraction, molality, relative activity or reciprocal absolute activity defining the pressure based, fugacity based, concentration based, amount fraction based, molality based, relative activity based or standard equilibrium constant (then denoted $K^\circ$ ), respectively.


The standard equilibrium constant is always unitless, as it is defined differently (goldbook)


  Standard Equilibrium Constant $K$, $K^\circ$ (Synonym: thermodynamic equilibrium constant)
  Quantity defined by $$K^\circ = \exp\left\{-\frac{\Delta_rG^\circ}{\mathcal{R}T}\right\}$$  where $\Delta_rG^\circ$  is the standard reaction Gibbs energy, $\mathcal{R}$  the gas constant and $T$  the thermodynamic temperature. Some chemists prefer the name thermodynamic equilibrium constant and the symbol $K$.


In your example both would work if you straighten your definitions, involving concentrations
\begin{aligned}
K_A \cdot c(\ce{X}) \cdot c^n(\ce{A}) &amp;= c(\ce{X_{A}}),\\
y=\frac{c(\ce{X_A})}{c(\ce{X})+c(\ce{X_{A}})} &amp;= \frac{K_A\cdot c^n(\ce{A})}{1 + K_A\cdot c^n(\ce{A})},
\end{aligned}

or activities
\begin{aligned}
K^\circ_A \cdot a(\ce{X}) \cdot a^n(\ce{A}) &amp;= a(\ce{X_{A}}),\\
y=\frac{a(\ce{X_A})}{a(\ce{X})+a(\ce{X_{A}})} &amp;= \frac{K^\circ_A\cdot a^n(\ce{A})}{1 + K^\circ_A\cdot a^n(\ce{A})}.
\end{aligned}

In both cases $y$ remains unitless.

In experimental chemistry it is much easier to observe concentrations instead of activities. Therefore to fist named constant is more often in use. A straightforward use of the thermodynamical equilibrium constant might sometimes prove a little tricky.

For reasonable dilutions one can simply substitute activity with a unitless concentration as given through
$$a(\ce{Y})=\gamma\frac{c(\ce{Y})}{c^\circ},$$
with $\gamma\approx1$ for $c(\ce{Y})\to0$ and $c^\circ=1\:\mathrm{mol/L}$.