I was dog sitting when the dog jumped my fence. The owner said the dog does that often at his house but comes back so there is a pattern. I was in the yard with 3 dogs the dog who ran away was around the corner I went to make sure she was in the yard and she was not! I put my dogs inside and ran to get her. I was outside for over an hour trying to find her the owner came and we found where the tracks ended. I told him to call the closest animal hospital and she was there, she was hit by a car. Am I responsible for the health costs?

Are you at the stage of Small Claims where you respond in writing to the complaint?
A lot will depend on the agreement you had and if it was in writing or not, whether this is something you do for others, whether it's a licensed business, etc. It will also depend on the judge and how you present yourself (appearance, manners, etc) in court, if you get to court.
The burden isn't on the dog owner to tell you what 'patterns' his dog has, although that would have been reasonable; it's on you to ask, just as you ask need to ask about shots and licenses and allergies and biting and so on, and write it all down.
If this was a one time dogsit akin to babysitting for a neighbor and not a routine venture for others too, the judge might take pity on you.
Small Claims go quickly and can be somewhat arbitrary, because it's not high stakes or criminal.