It's called "Jerry Gets Sick". Here's pbs.org's description:
Jim Nelson and Rene Agredano’s dog Jerry was the CFO (Canine Fun Officer) in the business they ran from their home office. But when Jerry became ill, Jim and Rene decided to sell their business and their house — and they hit the road with Jerry.
It's a must watch. If you don't come away from this segment without tears in your eye then you just ain't a dog lover. Click here to watch the full segment on the PBS Nature website.