The general boundary condition for the Dirac equation is a local linear restriction on the components of the spinor wave function at the boundary, $$\psi=M\psi,\;\;M=\begin{pmatrix} n_z&n_x-in_y\\ n_x+in_y&-n_z \end{pmatrix}$$ with ${\mathbf n}=(n_x,n_y,n_z)$ a unit vector. (Check that $M^2=\mathbb{1}$.)
If you are interested in the generalization of the Dirichlet boundary condition, then you will want to ensure that zero current flows through the boundary. This further restricts $M$ to vectors ${\mathbf n}$ that satisfy $({\mathbf n},{\mathbf n}_B)=0$, where ${\mathbf n}_B$ is a unit vector in the $x$-$y$ plane perpendicular to the boundary.
All of this was worked out in the context of graphene some years ago, see arXiv:0710.2723