How much social media control can your employer exert over you?

Question

Can a swiss employer force the employee to change the display picture and full name of your personal linkedin profile?

Answer 1

If your display picture or your user name have some clear reference to your employer then yes your employer may have a say in it. See stuart F's comment for examples, even if the examples are similar but positive about the employer the same would apply.

In all other cases it is none of your employers business what you do in your personal life and on social media (as long as it does not relate directly to your employer).