I double-checked recently because like I said, I'm starting my daughter (officially anyway) in a couple weeks and sure enough, nothing needs to be done. No notification of intent even. The only time anything comes into question is if the school district feels they have reason to believe that adequate schooling is not being provided for homeschooled kids. Even then, it is their burden of proof - they have to demonstrate evidence. NJ law can be found here.
My husband and I are from PA originally, and I have wanted for years to go back, still can't get used to the cost of housing here, among other things, and we're still too urban for my taste (we moved from NYC when I was pregnant with my daughter)...until I found out the homeschooling requirements in PA are MUCH, MUCH stricter than NJ. Suddenly NJ doesn't seem so bad after all. :)
That's wonderful about your boys!!!