If your friends damage something while helping you to move, their liability will be limited as they were doing you a favor. In this case, your friends' liability insurance will likely only cover a portion of the damage.
If you make use of a removal company, you should look at their general terms and conditions (GT&Cs) before entering into an agreement. Take a look at the amount for which the removal company is liable, because they may limit their liability. Clarify whether there are certain damages they cannot be held liable for. In any case, it is worth taking out special transport insurance.
Good to know: With Zurich your household contents are protected during the move up to a value of CHF 2,000, provided you have taken out the supplementary insurance "Accidental damage to contents".