Are 2x6 rafters adequate to carry a day bed in a pergola?
I think you'd be fine hanging the swing from both 2x6's.  (I don't see any reason you wouldn't use both.)  Southern pine 2x6's are OK up to 9'11&quot; for joists if spaced 12&quot; on centers. But those numbers are based on limiting the amount of deflection of the joist, so the floor isn't springy.  In this way, a little deflection isn't going to be a big deal, in fact a lot of those swings have a spring in the support so they have some give.
If you sink four eye bolts into the bottom of the 2x6's, two at each end, and put say 1' of chain on each eye bolt, and use a threaded link to connect that to the chains that the swing hangs from, it should be more than strong enough.