While there are a number of different reasons why a dog this age might cough, the most common reason, by far, is going to be Kennel Cough (aka Tracheobronchitis) since he was just recently boarded.
The barking that's he doing may exacerbate his coughing which could further irritate his throat but short of walking him on a leash when he's outside, there may not be much you'll be able to do that will prevent him from barking when he's out there.
Kennel cough is either secondary to a viral or bacterial infection and can be contagious between dogs. This is one reason why it's commonly seen where dogs spent a lot of time together such as doggie daycare, boarding or if groomed.
If this condition is secondary to a bacterial infection, then these dogs respond to oral antibiotics pretty quickly.
But if this is a virus , then it will have to run it's course....which is usually between 10-14 days.
Kennel cough tends to worsen before it improves and it typically gradually improves rather than abruptly stopping unless this is a bacterial infection and antibiotics are started.
These dogs also tend to cough more when excited, first thing in the morning or during the winter when they go outside in the cold (or come back inside where it's warm if they've been outdoors).
There is another condition called Canine influenza which mimics kennel cough in the early stages but these dogs rapidly become sick. Since Henry isn't acting terribly ill (aside from the coughing he's doing), I wouldn't expect this particular condition to be a problem but I did want to at least mention it.
Some kennels require Bordatella (kennel cough) vaccines prior to boarding which can reduce the incidence of this condition. But it doesn't 100% prevent it. This vaccine helps protect against the most common cause of Tracheobronchitis but won't be effective against viral causes in case you were wondering.
As to over the counter treatment options, cough suppressants can be given although I'm often hesitant to use them if kennel cough is the problem. I want those secretions removed from the upper airways and I rarely want to inhibit this reflex.
But acceptable ones to use include Dextromethorphan. The dose would be 0.25 to 1 mgs per pound 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.
I'll sometimes suggest their use if the patient is coughing so much at night that no one in the household is getting any sleep, though.
I'd also suggest that you feed softer foods such as canned or semi-moist so as not to further irritate the throat with hard kibble. Alternatively, you can soak kibble in water for 15-20 minutes which will soften it up quite a bit.
If he develops other symptoms such as lethargy or disinterest in food or a yellow/green nasal discharge, then a vet visit may be prudent. Otherwise, there's nothing wrong with continuing to monitor him and wait for the condition to run its course.
I hope this helps. Deb