mechanical resonators. The proposed hybrid circuit architecture comprises a superconducting transmon qubit coupled to a pair of magnonic Yttrium Iron Garnet (YIG) spherical resonators or mechanical beam resonators via flux-mediated interactions. Specifically, the coupling results from the magnetic/mechanical quantum fluctuations modulating the qubit inductor, formed by a superconducting quantum interference device (SQUID). We show that the resulting radiation-pressure interaction of the qubit with each mode, can be employed to generate maximally-entangled states of magnons or phonons. In addition, we numerically demonstrate a protocol for the preparation of magnonic and mechanical Bell states with high fidelity including realistic dissipation mechanisms. Furthermore, we have devised a scheme for reading out the prepared states using standard qubit control and resonator field displacements. Our work demonstrates an alternative platform for quantum information using ECS in hybrid magnonic and mechanical quantum networks.