This is bad for several reasons. First, and most obviously, we'd have to pay for it. BP caused this disaster, and they should be responsible for cleaning it up. If they are run out of business, they can't. I have to buy gas for my car either way, so if buying BP gas makes it less likely that I'll have to pay for cleaning the Gulf, then I'll do that. Secondly, a government cleanup would be less efficient than a BP managed one. BP currently has access to the best of their own engineers and the best of the government's. If BP goes away, then all of the BP engineers go away. Unless there are no BP engineers contributing anything of worth at the moment (which seems unlikely), this would be a bad thing for the gulf coast. A government run cleanup would also be less efficient because of the incentive distortion. Right now, BP is the one running the cleanup AND the one paying for it. If the government were running it, then it would be government agents running the cleanup and taxpayers paying for it. Simple economic theory tells us that when actors are shielded from the costs of their actions (in this case, the cost of running the cleanup inefficiently), they will not perform at the socially optimal level.
In conclusion, while boycotting BP is unlikely to have much effect at all, it is my belief that it can only have a negative effect. I'm assuming that the goal of the boycott is to force BP out of business, which I've shown is bad for both the gulf coast region and for taxpayers. As far as the implied benefit by removing a reckless company from the market, the best way to ensure another oil spill doesn't happen is to make government regulation more effective and efficient. BP is unlikely to survive another catastrophe like this, and they know it. As long as government oversight is improved, BP is no more of a threat to our environment than any other oil company.