differential equations - Eigenvalue dependent boundary conditions- mathematica

I am dealing with an eigenvalue problem whose boundary conditions are also eigenvalue dependent.

Could anyone please comment whether Mathematica can numerically solve such a problem? For boundary condition independent of eigenvalues, I use NDEigenSystem.

A minimal working example is given here. The eigenvalue problem: $$ -\frac{d^2 \psi}{dx^2} +x^2 \psi = E \psi $$ with two boundary conditions: $$\textrm{(i) }\psi = 0 \textrm{ at }x = 0$$ and $$\textrm{(ii) }\frac{d\psi}{dx}+E^2\psi = 0 \textrm{ at } x = 1$$ needs to be solved to calculate the eigenvalues, $E$, of this operator. This might seem to be a trivial task, but please be aware of the eigenvalue-dependent boundary condition. I would be very thankful if anybody could suggest how to solve such eigenvalue problem in mathematica.