Blackie refuses to put his leash on and go for his walk. His owners are sticklers for him not peeing in the house. Chasing him around the house hasn't worked. Offering a treat hasn't worked. He got out of his crate, and no way does he want to go outside. What are some good counter-actions on the sitter's part to this hide and seek behavior?
What size is Blackie? If he's small enough to pick up, I would suggest just picking him up, putting the leash on and carrying him outside. Otherwise, I don't have a clue.
I have to do that every so often with my dog, usually when it is raining. I carry him down my very steep stairs and then he'll just stand there and won't move. It can be a struggle.
My younger dog hates to go outside if it's wet/cold/windy (I never thought I'd have such a prissy dog, ha!). She usually retreats to her kennel and I have to crawl in after to clip her leash on to lead her out. But in your case it sounds like he's running away from you? I'd try to corral him into a smaller space, like a bedroom or bathroom so he doesn't have so far to go and it will be easier to catch him. making a slip lead out of his leash (run the length of the leash through the handle to make a big loop) so you can kind of lasso him might be helpful, too. If you can't catch him, notify the owners and Rover of the situation.
Are you travel sitting or just giving mid-day walks while the owners work? If it's the former, I'm concerned that it sounds like this dog is spending all his time in the crate except for walks while his owners are away.