I've never heard it before, certainly not as some kind of common phrase - but it sounds to me like the sort of line you'd hear in a cheesy wartime spy movie.  Some sort of code phrase to inform your accomplices of your plans.

Origin and meaning of "The eagle flies at midnight"
Assume  that $H$ is  a  separable Hilbert space.
Is there a polynomial $p(z)\in \mathbb{C}[x]$ with $deg(p)&gt;1$ with the following property?:

Every densely defined operator $A:D(A)\to D(A),\;D(A)\subset H$  with $p(A)=0$ is  necessarily  a  bounded operator on $H$.

That is the polynomial-operator equation $p(A)=0$ has only bounded solution.
