I must be the exception to the rule, ZigZag as I don't feel unhappy, insecure, or resentful, because I didn't have a 'conventional' childhood living in one place and only attending 1 or 2 schools before leaving home to live with strangers at university.
I got moved around a lot as a child, and attended a considerable number of day and boarding schools in 5 different countries on both sides of the pond. My education didn't suffer as I was always able to make the necessary transitions (at some schools I was way ahead of my class, and at others I had shift my ass to catch up) and get good grades.
When I was very young I did find it hard to say goodbye to friends and familiar routines before venturing into the unknown - however, I soon learned to welcome change and began to get a tad bored if I had to stay in one place longer than a year or two.
Looking back, I believe that my nomadic childhood made me more socially capable and self-reliant at an earlier age than my peers, and the wealth of experience that I'd accumulated by osmosis stood me in good stead when I went to college a year earlier than my peers.
A big plus is that my early travels have resulted in my having a number of long established friends in Europe and the US that I enjoy visiting and hosting and we, and our dcs and other assorted friends/family members, all benefit from relatively cheap holidays/breaks without extensive planning - to say nothing of a wealth of local knowledge on tap for guidance to the best places to visit/eat/shop/play/rent cars/change money etc.
It's my understanding that educational standards in Bermuda are high, but you can seek advice from your dc's present school(s) if you have any concern that they may fall behind their peer groups.
Will the year in Bermuda conclude with your dcs returning to their present school(s)? If so, it's likely that their friends will be green with envy as am I, and as it's only a 2hr or so flight away from NYC maybe they could have their special pals over for vacations and/or stay in touch via skype or similar?
One school year is short, and so is life - go for it!! The only downside I can foresee is that you may not want to leave such a beautiful island.