Well theres a turn up for the books mate. As you know, you can stay in Corfu for as long as you like, being part of the EU, but you must claim residency after 3 months. There has been many topics on this for the details of what to do, most recently from Debbie as she is moving out next month.
As far as National Insurance, whats wrong with paying it yourself direct from your own funds. I did this when I was unemployed. I was entitled to any benefits because I get a Naval Pension, so I could not be bothered to got to the Job Centre every week or so, so I contacted the appropriate offices and they said I could pay myself through DD. I did this until I started work again. I think you will find that you have to have 44 years paid NI to get the full pension, if one is behind, you can play catch up to meet your contributions. I do not see any reason why you cannot do that from abroad. We have seen and heard people talking of having their OAPs paid into their UK banks, and drawing it from ATMs or have so much transferred to their Greek Bank.
Basically, upto to 3 months, no problems, 3 months and more you need a residency permit to remain in Corfu.
No doubt someone more in the know will add to this.