The
Gulf Stream can be a quandary for anyone planning to go to
Bermuda from most any port on the eastern seaboard.
The location of the Gulf Stream and it's various eddies change almost daily...so for a June trip, not much point in analyzing today's GS location.
The big bugaboo about the GS is when you when you have winds blowing against the axis of the stream, which tend to build steep waves for a period of time.
You can
google "Mike's
Weather Page" , which contains
current GS location, meanders, etc, plus a variety of other
charts depicting wave heights and so on to get a feel of what you are likely to have to deal with.
Additionally, prior to your trip you can contact folk that specialize in
weather routing, that will advise when and where to go, keeping in mind the location of the GS.
From a practical point of view, having done the trip to
Bermuda several times, though not from Boston, you can also use the GS to your advantage, as stated above, if you know it's location and its meanderings.
Getting there is one thing, but you will also need to get back, which depending on your length of stay, may put you in July or later, sometimes requiring a detour to an alternate port of entry, which has happened to me, but that is sailing for you, ie, to get from A to B, sometimes requires going to C first.
There are also several races to Bermuda each year, which you could join, which might give you some comfort, though after the start, it's unlikely to see another competitor again until you reach Bermuda.
In June, you can expect relatively settled weather, but the odd anomaly could still occur. I had my ears clipped one year returning from Bermuda in July.
My final piece of
advice. Bring a fat wallet. Bermuda tends to run on the expensive side for a visitor but it's a cool place to visit.