Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$ on such an $M$ with Dirichlet or Neumann boundary conditions. In such situations, are the eigenfunctions of the Laplacian smooth? It seems to me that this requires some sort of elliptic regularity theory for singular spaces (maybe there is an elliptic regularity theory for manifolds with Lipschitz boundaries). A reference would be highly appreciated.

There is an extensive literature on elliptic problems on domains with corners. Grisvard's book (Elliptic Problems in Nonsmooth Domains) is a good place to start.

Let me add some other references about elliptic regularity on singular manifolds:

1) B.-W. Schulze and his co-authors have developed a comprehensive calculus for elliptic pseudo-differential operators (in particular differential operators) on singular spaces, including elliptic regularity and Fredholm properties:

1.1) For manifolds with conical singularities and manifolds with edges consider the following books:

B.-W. Schulze. Boundary Value Problems and Singular Pseudo-Differential Operators. J. Wiley, Chichester, 1998.

Ju.V. Egorov and B.-W. Schulze. Pseudo-Differential Operators, Singularities, Applications. Birkhäuser Verlag, Basel, 1997

1.2) For manifold with corners consider the following paper:

Schulze, Bert-Wolfgang. "The Mellin pseudo-differential calculus on manifolds with corners." Symposium “Analysis on Manifolds with Singularities”, Breitenbrunn 1990. Vieweg+ Teubner Verlag, 1992.

1.3) For manifolds with corner and edges the following is an upgraded version of the last paper:

B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics, Publications of RIMS, Kyoto University {38}, 4 (2002), 735-802

1.4) For a more geometric treatment of elliptic operator on singular spaces consider:

Nazaikinskii, V. E., Savin, A. Y., Schulze, B. W., & Sternin, B. Y. (2005). Elliptic theory on singular manifolds. CRC Press.

In these papers/books they approach the problem with the goal of obtaining an algebra that contains elliptic operators and their parametrices (inverses module compact operators). If you are interested in a particular class of operators and need a concrete theory for that specific class you should also check the books of Mazya/Rossman/Kozlov:

Mazya, V. G., and J. Rossmann. Elliptic equations in polyhedral domains. No. 162. American Mathematical Soc., 2010. (this book might be particularly useful for the Laplacian on a cube)

Also for eigenvalues and spectral theory check this one:

Kozlov, Vladimir, Vladimir G. Mazí̂à, and Jürgen Rossmann. Spectral problems associated with corner singularities of solutions to elliptic equations. No. 85. American Mathematical Soc., 2001.