What do you do if you are pet sitting/overnighting at the owners house and one of their pets eats/ruins some of your things?

There's nothing you can do except grin and bear it. Next time, you'll need to take more precautions about your personal belongings when there's a client dog on the loose. Personally, I wouldn't mention it to the client, except maybe in passing. Definitely don't ask or strongly suggest they reimburse you.

You could include the replacement as a business expense for tax purposes. ;-)

BTW, I realize you were staying at the clients. However, at my own home, I have about a dozen sheos/boots near the front door. I always ask clients whether they are safe. I've never had an incident of a client dog destroying anything in my home.

You can always ask the client to compensate you for the damage, but I wouldn't expect to be compensated. Damage to your home and personal belongings are just part of the business. And an incident like this makes you think more about protecting your home and placing anything of value out of the way of any visit dog.