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Of course they can - for themselves; just as everyone else is free to do. At least, in theory. But not in practice, of course. That is decided by the STATE.
My issue with the BNP, as with the liberal-left, is both their attempts to impose their own definition (or cosmopolitan lack of it) on everyone else, via the state, on which we all depend, and seek to influence to our own perceived (economic, social, political and/or ideological) advantage.
A healthy and humane society is one whose members freely identify with it as their OWN (constituting a genuine PEOPLE and NATION, the natural and legitimate heir of one's original tribe), but the STATE long ago took that choice away from us, coercing us with a system of (promised) rewards and (threatened) punishments (just as we train dogs) to identify with itself, however defined.
The Nazi fascist state brutally imposed its own, insane, definition of a mono-ethnic identity, while the liberal-fascist state (see Jonah Goldberg's book "Liberal Fascism") imposes a multi-ethnic melting pot on our society, in which the human diversity it claims to love, must in the longer term surely dissolve and disappear (most human diversity being a consequence of human populations having been effectively isolated from each other in the past).
What I would like to see a truly liberal and enlightened (as opposed to liberal-fascist, i.e. statist) state do, is stop pretending, and enforcing the ideology that "ethnicity doesn't matter", i.e. is of no social or political importance except to evil "racists", and allow its citizens to freely choose - peacefully and grassroots-democratically - their own national identities.
Britain and America would then become multinational states, which, I am convinced, is the way to create the kind of just, humane and sustainable societies (note the plural) most liberal-lefties actually want, but mistakenly believe only the STATE can provide.
What the STATE wants - as presently constituted, above all else - is POWER.