I just dug up information from the lowes site on the turned wire type eye bolts.
The 3/8ths is 190 the 1/2 is 300 and the 5/8 was 560 pounds.
Stanly brand.
The forged eye bolts go up into the 2000 pound range or more.
Remember the two bolts that connect to the swing will share the load but they will be at an angle.
So 190 X 4 is 760 divided by 3 is 253.3 X 2 = 506.6 pounds.
Jethro math but it works for me.
This is accommodating the side load on the eye bolts at a 60 degree angle.
So a 3/8 bolt would be fine.
The top eye bolts I think would be fine at 300 pounds each also.
If it were me I would run both chains on both sides to the very top and get rid of almost all of the side load and connect them to the link right under the eye bolt there.
This will also keep the seat from flipping and relieve a ton of stress on the wooden seat.
Worth