I don't think any business would allow you to pay "when you get the money" that could possibly be months down the road...I'm sure they've had many people just not pay them. That really doesn't seem ridiculous to me, honestly. It's sad they would put him to sleep after a week though, I hope that your friend can figure out a way to get Milo back. This is why it's so important to have an emergency fund for your pets.

Could they see about setting up a payment plan? If the vet does not do that, they could put it on a credit card and pay it off a bit at a time until they have more money (since she's supposed to have more money in December anyway).