I'm not quite sure how you are getting an eigenfunction problem or what you expect the eigenfunctions look like. Can you elaborate? Ignoring the $x(c)=0$ condition momentarily, we can easily show that the solution to this ODE is $x(t) = a\sin(t\sqrt{\frac{k}{m}})$ with $a$ free. Adding the $x(c)=0$ condition back in either forces $a=0$ (if $c$ isn't a zero of the sine wave) or tells us nothing.

You can view this particular problem as finding an eigenfunction with eigenvalue $-k$ for the linear differential operator $m\frac{d^2}{dt^2}$ which satisfies the conditions. I don't think this is what you're referring to, but I'm not sure what you are referring to.

Just suppose we solve this problem as boundary condition problem,not by just solving the ODE in the "simplest" way

There is not a "singular" way of solving a boundary condition problem, so saying "solve this problem as a boundary condition problem" doesn't specify an approach. Nevertheless, I assume you mean something like this. This corresponds to my second paragraph; we're just straight-up noticing that the problem is of the form of an eigenvalue problem for a differential operator. If this is what you had in mind, then I don't understand your question.

The eigenfunctions are the solutions to the unconstrained problem and the eigenvalues have whatever physical significance they have in the problem. So the "physical meaning" is the same as the "physical meaning" of these things in the problem. For this example, the eigenfunction will be the trajectory of the mass and the eigenvalue is the spring constant. The physical meaning of the solution doesn't change based on how you found it.