As dmckee says in his comment, the answer is no, a stationary spherical shell isn't possible. This is because not even the interparticle forces in neutronium are strong enough to support it.
The problem is that once inside the event horizon there is no way to travel away from the singularity, or even maintain your distance from it, without travelling faster than light. However the fundamental forces like electromagnetism and the weak and strong forces have a maximum propagation speed of $c$, and that means the three forces cannot propagate outwards. A neutron at the outer edge of the spherical shell can't be held there because none of the three forces can move outwards to hold it there, so it must fall inwards. This applies to all the neutrons in the shell, so they and the shell must inevitably fall inwards and hit the singularity.
Proving this is a bit involved. see my answer to Why is a black hole black?, but even though I've tried to simplify the maths it's still a bit intimidating.
I've just spotted Can we have a black hole without a singularity?, which is not exactly a duplicate but is related.
You might have noticed that dmckee said in his comment:
No more than a strong spherical shell elsewhere could resist time passing
and you might wonder what this means. Well when we talk about objects falling into black holes we generally (often without realising it) choose a set of co-ordinates called Schwarzschild co-ordinates. These are time, $t$, radial distance from the centre of the black hole, $r$, and two angular coordinates that I'll ignore for now.
As long as you're outside the event horizon $t$ and $r$ make intuitive sense. It's true that time slows down for an object near the event horizon, but we're already used to time changing for fast moving objects. Time still behaves like time and distance still behaves like distance. Specifically, we can move in either $r$ direction but we can only move forward in the $t$ direction i.e. we can't move back in time, only forward.
However, if we use our Schwarzschild coordinate system to describe events inside the event horizon we find something very odd. The $r$ coordinate that describes distance from the centre of the black hole now behaves like time, and just like time it's now only possible to move forward (towards the singularity) in $r$ and it's impossible to move backwards. This is why dmckee made the reference to the shell being unable to resist time passing. The spherical shell inside the event horizon cannot resist moving forward in $r$ any more than a spherical shell outside the event horizon can resist moving forward in $t$.
This sounds weird (it's used over and over in science fiction), but it's perfectly good physics. However I'm not fond of this viewpoint because the apparent weirdness is the result of a poor choice of coordinates. Since in the Schwarzschild coordinates an object takes infinite time to reach the event horizon you shouldn't be surprised to find the Schwarzschild coordinates do a poor job of describing what happens inside the event horizon. It's important to emphasise that if you were falling into a black hole you wouldn't experience distance behaving like time or vice versa. In your (brief!) journey to destruction at the singularity you'd find time and distance behaving just as you'd expect them to.