The answer to your question is negative. I have to be brief, so I will only give an outline. I will try to expand on it tomorrow.
In the entire case, it is possible to construct an entire function with the following property: The Fatou set consists of a single conneted attracting basin (hence the nonescaping set is connected), the non-escaping part of the Julia set contains no curves (actually, has Hausdorff dimension zero), and there is a non-escaping point in the Julia set that is not accessible from the Fatou set.
This shows that the non-escaping set is not path-wise connected, and hence not contractible. (The construction is contained in an upcoming article of mine, dealing more generally with the topology of transcendental Julia sets.)
For quadratic Cremer polynomials, the key point is that the Cremer point is accumulated on by small cycles by work of Yoccoz, and some of the arcs (if any) connecting these to the fixed point must be quite long.