As a preamble to my answer I almost feel obliged to say I have never voted for the BNP and can never envisage doing so .. that should not even be necessary but in these political correct days it is often difficult to hold honest views about a matter or be open minded without feeling one has to defend a standpoint .. so here goes.
On the basis the BNP is a lawful, recognised organisation, that the officer was off duty, that he was not participating in any political demonstration or similar I do not feel he should have been required to resign. I say this much on the basis that 'whilst I dislike what you say as long as it is not unlawful I defend your right to say it'.
I very much doubt that the Chief Constables force orders are even legal in themselves and could well be struck down as 'ultra vires' if a legal challlenge was mounted against them, at least as far as actual membership of the BNP itself is concerned. Banning membership seems a very slippery slope on the way to saying who a police officer can or can't support politically or vote for. Additionally the orders came into being long after the officer joined the force so the arguements about it being a specific term of employment doesn't come into it.
What has happened is the force has chosen to hold an inquiry and required the officer to resign. this on the police scale of punishments is one down from actually dismissing him themsleves. The officer has the right of appeal to the Home Secretary and I hope and expect he may do so. The officer could also take this to the European courts.
As others have pointed out, some of the BNP policies, weekly bin collections etc. are very live, local, man in the street issues, could it be that a police officer in the BNP with views and training of fairness just might make the BNP more credible and less racist ? No one appears to have accused or found the officer to actually be racist so why make that presumption in the absence of evidence. Is he not innocent until proven guilty which is what you expect of the man himself ?
I do declare an interest in this as a retd member of a UK police force ..others must judge the impartiality or fairness of my A. I assume the facts are as reported and have no inside information about the incident.