Recent experimental work has demonstrated nano- textured magnetic Josephson junctions (MJJs) that exhibit tunable spiking behavior with ultra-low training energies in the attojoule range. MJJ devices integrated with standard single-flux-quantum neural systems form a new class of neuromorphic technologies that have spiking energies between attojoules and zeptojoules, operation frequencies up to 100 GHz, and nanoscale plasticity. Here, we present the design of neural cells utilizing MJJs that form the basic elements in multilayer perception and convolutional networks. We present SPICE models, using experimentally derived Verilog A models for MJJs, to assess the performance of these cells in simple neural network structures. Modeling results indicate that the tunable Josephson critical current IC can function as a weight in a neural network. Using SPICE we model a fully connected two layer network with 9 inputs and 3 outputs.