We've done both...
Either month has its own pro's and con's..
June: Less chance of bugs, but less chance to catch turtle nesting..
July: More chance of bugs, but again better chance of turtle nesting happening..
Personally, the bugs have never been a big deal for us... (We got bit up a little last year from sand fleas on the beach, but that wasn't a big deal...) Most of the time, the "no see-ums" seem to leave us alone.