People born in Anguilla automatically hold British Overseas Territories citizenship, which entitles them to a British passport but not to the right to live in the UK.
However anyone who held such citizenship prior to the British Overseas Territories Act 2002 coming into force also holds full British citizenship as does any other person from Anguilla who has made an application for such citizenship and when the Secretary of State has then granted it.
As a full British citizen, a former resident of Anguilla would then become entitled to the same rights as someone born in the UK.
The status of British Overseas Territories is somewhat complicated, as they're ruled in many ways by the UK Parliament but don't have any seats in it. (Both the Lib Dems and UKIP have called for them to be given seats in the House of Commons).