True, however differential equations inherently describe physical phenomena right? That is why initial or boundary conditions make sense. Also, keep in mind that in the general case (for a random 2nd order ODE), if you just write it down in Sturm-Liouville formulation, the eigenvalue will be 0. It is non-zero only if:
1. The problem the equation describes has this property.
2. Even if it does, the boundary conditions must be such that the eigenvalue is not zero.
But from a more abstract standpoint, although I may not be the most qualified person to discuss it, eigenvalues are simply parameters. Take the Helmholtz equation:
What this tells you, is that for a given set of boundary conditions, you can have various (often infinite) solutions. Practically, as many as the possible values of λ. That is why these are called eigenvalues (=values of self), they are an intrinsic property of the equation (or, by extension, physical problem) itself, and not dependent on the value of either x or y.
Edit: To help visualise it a little better, eigenvalue problems arise often in wave equations, where the eigenvalue relates to the period (or frequency) of the wave.