Greetins, islandliving, welcome to the forum.   

Good luck with the new gig in Bermuda... keep us posted how 'savy' you feel after a year there!

Here is the info regarding Social Insurance:  ($59.80 combined from you and your employer, every week)

Contributory Pension

You are entitled to a Contributory Pension if you:

are 65 or over; and
have paid or been credited with a specified number of contributions and the average number of contributions paid and/or credited is between 25 and 50 a year.
The amount you receive will depend on the total contributions and number of years you have been part of the scheme.

You may qualify whether you stop work or not.

Non-Contributory Pension

You qualify for a Non-Contributory Pension if you:

do not qualify for a Contributory Pension; and
are 65 and living in Bermuda; and
are Bermudian and have been ordinarily resident here for a period of not less than 10 years during the 20 years before you apply.

Bermuda has no tax treaty with the US.  If you're self-employed, you'll have to pay 13.2% SSI on the first ~100K of your income.  If you're not self-employed, you don't have to pay SSI.  At least this is what my tax advisor has told me, so if he's wrong, he can go explain it to the IRS. 

If you're working for a Bermuda company (not the Bermuda subsidiary of a US company), you should be exempt from SSI.  Bermuda has some payroll tax, which I think is a half/half thing, so that might be what your employer is referring to.  (Maybe someone can correct me there.)

It is worth consulting a tax advisor who specializes in expat taxes before coming down here to estimate your tax burden and set-up your quarterly payments.  If you don't play your cards right, you may find that the US or your state claim you're not a bona-fide foreign resident, and you lose your state tax exemption and the $100K/person expat exemption from Federal tax.  Also, taxation of things like housing and travel allowances gets devilishly complex.   

There is also a Bermuda social security payment of around $100 per month which is usually split 50/50 between employer and employee.