Keep them away. Sorry, I don't want a little pussy or puppy eyes watching my HE. And the hair thing would have sent me out the door. Given the fact that is common for the nasal passages to loosen during a massage, especially face down, and the amount of natural toxins being loosened in your body during the massage, the last thing anyone needs is a potential allergen.
Now, I'm not a Felix Unger, Sheldon, or Seinfeld when it comes to germs, but....
Pets should be seen and not heard, and if there is one, the masseur should have the courtesy to mention it at sometime. And then be absolutely fn impeccable about it.
... rant over...