Hello, I'm Dr. Deb and will do my best to help you today.
I'm sorry for this concern for your dog. While the ingestion of such things like a chicken bone can certainly cause coughing secondary to initial irritation when it was ingested, it seems a little less likely that you'd be seeing signs two days later....perhaps a delay of 24 hours or so but 48 hours seems like a long time to me for this to be the problem.
While there's a fairly long list of reasons why a dog might cough (ranging from bronchitis to collapsing trachea to cardiopulmonary disease), the most common cause of coughing in a dog this age is going to be tracheobronchitis aka kennel cough. Even though he may not have exposure to other dogs, we can bring home pathogens on our clothes and shoes and expose our pets.
Additionally, the vaccine he received only protects him against the most common bacterial cause of this condition. It's a common misconception that he's completely protected since he could develop symptoms if exposed to a virus. And, if this is a viral infection, it will have to run its course which may take 7-10 days.
As long as he's otherwise active and eating, I wouldn't consider this to be an ER type of situation; you can continue to monitor him for a few days.
But I do have a few suggestions which may be of help:
1. Feed kibble soaked in water for 15-20 minutes or canned food so that the throat isn't irritated further.
2. Over the counter cough suppressants can be given to dogs although I'm sometimes hesitant to use them if I suspect Kennel Cough.
I want those secretions removed from the upper airways and I rarely want to inhibit this reflex. But acceptable ones to use include Dextromethorphan: http://www.petplace.com/article/drug-library/library/over-the-counter/dextromethorphan-robitussin-dm
The dose would be 0.25 to 1 mg/lb 2-3 times a day. You just want to double check labels and ensure that the formulations only contain this ingredient although inclusion of Guaifensin is fine.
If one of your other dogs starts to cough within another day or so, then you'll know that you have a contagious condition. But, even if he remains the only symptomatic patient, if he continues to cough then a vet visit might be prudent so that his heart and lungs can be evaluated since there are other reasons why a dog might acutely start coughing.
I hope this helps and that you see improvement soon. Deb