You can use a Cloth simulation with Internal Springs active and Pressure set to a decently high number (1000 tends to do it but experimenting can help you find a better result). You can limit this simulation to only a few select vertices with vertex groups (in Shape -> Pin Group)

An alternate method is to use a cage mesh that covers the area you want to add squish physics to, then using a mesh deform modifier on the body to copy the deformations. I don't recommend the latter in this case because on smaller models it offers negligible performance gains for extra hassle.