We investigate the spectral statistics of a network model for a three dimensional layered quantum Hall system numerically. The scaling of the quantity J_0=1/2< s^2> is used to determine the critical exponent ν for several interlayer coupling strengths. Furthermore, we determine the level spacing distribution P(s) as well as the spectral compressibility χ at criticality. We show that the tail of P(s) decays as (-κ s) with κ=1/(2χ) and also numerically verify the equation χ=(d-D_2)/(2d), where

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... is the correlation dimension and d=3 the spatial dimension.