How to best barbecue calamari/squid?
See:  How do you cook calamari / squid and avoid making it tough?

The principles remain the same for grilling or barbecuing:  you want to cook the squid briefly so it doesn't toughen up, or slowly for a long time, so that the collagen dissolves to gelatin, making the squid tender again.  In between is the rubbery zone.

The main issue is that many squid are not really large enough to place directly on a grill grate without danger of falling through, so you may wish to use a fish or vegetable grilling grate.  You can then grill them briefly, over high heat.

It is probably possible to use low-and-slow smoking techniques (true barbecuing), but I didn't find any really credible sources or recipes, and I have never tried this myself.  You could also grill the squid to get some smokey flavor, then finish by braising them (for example, in a disposable pie tin on the cool side of your grill, with flavorful liquid) for a long time.

Googling barbecued squid recipe will give you many recipes to try, with a variety of marinades and treatments.