Is there a field equation which can reduce into all three flavors of spin (zero, one, one half)? Is there a known particle field equation of a similar form 
$$
\begin{equation}
(\Gamma^n \pi_n)^2 \Psi = (mc)^2 \Psi \tag{1}
\end{equation}
$$
such that by reducing the number of degrees of freedom for the spinor $\Psi$ into a spinor of lesser degrees of freedom, such as a scalar $\psi_0$, two three-vectors $\boldsymbol{\psi}_\pm$ or two two-vectors $\boldsymbol{\phi}_\pm$, it reduces Eq. 1 into either ...


*

*a spin zero field equation 
$$
\begin{equation}
\pi^n \pi_n \psi_0 = (mc)^2 \psi_0, \tag{2}
\end{equation}
$$

*a spin one field equation
$$
\begin{equation}
(I\pi_0\pm i \boldsymbol{\pi} \times)
(I\pi_0\mp i \boldsymbol{\pi} \times)
\boldsymbol{\psi}_
\pm = (mc)^2 \boldsymbol{\psi}_
\pm \tag{3}
\end{equation}
$$

*or a spin 1/2 field equation
$$
\begin{equation}
(I\pi_0\pm\boldsymbol{\sigma}\cdot\boldsymbol{\pi})
(I\pi_0\mp\boldsymbol{\sigma}\cdot\boldsymbol{\pi})
\boldsymbol{\phi}_\pm = (mc)^2 \boldsymbol{\phi}_\pm? \tag{4}
\end{equation}
$$


In these expressions $\pi_n$ is the four-component momentum operator which includes the electromagnetic four-potential interaction $A_n$ with the particle's charge $q$ written as
$$
\begin{equation}
\pi_n = i\hbar \partial_n - q A_n , \tag{5}
\end{equation}
$$
and
$$
\begin{equation}
\boldsymbol{\pi} = -i\hbar \boldsymbol{\nabla} - q \boldsymbol{A} \tag{6}
\end{equation}
$$
uses bold to indicate a euclidean vector, specific to 3-components. The three two-by-two matrices $\boldsymbol{\sigma}$ in Eq. 4 are the Pauli Spin Matrices.
 A: The known wavefunctions for scalar, spin-1/2 and vector fields follow from the theory of unitary representations of the Poincare group. Especially the theory of induced representations (keyword here is Mackey's machine) and are special cases of a general wavefunction that an be written in the form 
$$Q(p)\psi = \psi$$
where $Q(p)$ is a projection operator.
Given a subgroup $K$ of a lie group $G$ we can induce a unitary rep of $G$ via a unitary rep of $K$ (also called the 'little group') by the following: Take a more or less arbitrary (non-unitary) rep of $G$, $D(g)$ whose restriction to $K$, $D(k)$, is unitary. We then construct a unitary rep of $G$, U(g), with
$$U(g)\psi(c) = D(g)\psi(cg)$$
where $c\in C = G/K$ is an element of the right coset space.
Now for the Lorentz/Poincare case the little group is $SU(2)$ and the coset space are the boosts, i.e. momenta. Therefore the above tells us that given a finite dimensional rep of the Lorentz group that is unitary on $SU(2)$, we can have a continuous unitary rep of the Lorentz group, which is the field $\psi(p)$ and that must obey the subsidiary condition
$$Q(p)\psi(p) = 0$$
(this is what will be called the wave equation)
and transformation law 
$$(U(\Lambda)\psi)(p) = \psi'(p) = D(g)\psi(p') = D(g) \psi(\Lambda^{-1}p)$$
the projectors $Q(p)$ for the common cases are:
scalar: trivial $\psi'(p) = \psi(\Lambda^{-1}p), p^2\psi = m^2\psi$ (KG)
spin $1/2$: $Q(p) = \frac{1}{2m}(\gamma^\mu p_\mu + m)$,  $(\gamma^\mu p_\mu-m)\psi = 0$ (dirac)
vector: $Q(p) = g^\mu_\nu - \frac{p^\mu p\nu}{m^2}$,  $p_\mu A^\mu = 0$(proca)
For a generic spin $j$ we get the Bargmann Wigner equation
$$(\gamma^\mu p_\mu - m)_{\alpha_r\alpha'_r}\psi_{\alpha_1\dots\alpha_r\dots,\alpha_{2j}} = 0, r=1\dots2n$$
A good reference on this 
Niederer, U. H. and O'Raifeartaigh, L.
Realizations of the Unitary Representations of the Inhomogeneous Space-Time Groups II. Covariant Realizations of the Poincaré Group
parts I and II
