The following is a slightly edited re-post from the Comment section of a March 2009 post entitled "My least favorite time of year":
June is dreadfully slow when compared to the madness that is May, so most people take a break from birding during that month. But the track record for rarities in June is quite impressive.
Sandwich and Gull-billed Terns, each with only a handful of accepted records in CT, may appear in June. Wilson's Plover is another target, along with Black-necked Stilt. Black-bellied Whistling-Duck has shown a recent June pattern in the northeast. Arctic Terns, particularly first-summer birds, seem to 'peak' in the region in June. Mississippi Kite's best time is early June, although we may now have a breeding population in New England! Old World shorebirds are certainly possible, particularly Curlew Sand and Ruff. Fork-tailed Flycatcher seems possible anytime, June included.
Also throw into the mix the presumably failed breeders that sometimes wander their way to our area. Chestnut-collared Longspur is a fine example of this. Scissor-tailed Flycatcher and Yellow-headed Blackbird too.
The southern breeders on that list are undoubtedly more common just across the sound on LI, some much more so (for example, Gull-billed Tern breeds on Long Island).
I'm probably forgetting a few species, but those are the ones that come to mind.
June is definitely an underrated birding month. It can be very productive, especially along the coast. Most people are just so exhausted from their full-tilt May birding that they don't get out much in June.
If anybody else can think of other June rarities, go ahead and post a comment.