It is highly unlikely that a ground squirrel digging around the root zone of an established or mature tree will cause any notable damage.  
If the tree is newly planted (in the last 3 to 5 years), there is a higher chance of causing problems.  The root systems of newly planted trees are much more limited.  This means that soil disturbance around the roots can cause them to potentially dry out more quickly, reducing the number of viable roots for a tree already low on roots.  Ground squirrels tend to be attracted to recently disturbed soils, making newly planted trees are more likely target for digging.  

The best method for discouraging digging would be to exclude the ground squirrels with hardware cloth.  Replace the missing soil and then cover the ground around the tree with hardware cloth.  Be sure the area protected is broad enough to protect the entire root zone of the tree.  There are other methods that can be used for control.  More information can be found in the link below
https://icwdm.org/species/rodents/thirteen-lined-ground-squirrels/thirteen-lined-ground-squirrel-prevention-and-control-methods/