So while I agree with most of what has been posted, I thought I might throw in my two cents as the voice of reason here.
Yes, in most states, you have to have a hunting license to hunt AT ALL. Song birds are also protected in pretty much any state that I'm aware of. BB guns are not necessarily approved "legal methods of take" in many states, but are rather considered to be firearms, and regulated for hunting as such.
NOW, on the other hand....
If you're talking about just dinking around in your backyard and occasionally popping some BB's after a few squirrels, then you'll likely fall under the "no harm no foul" mentality. Sure, it may not meet the letter of the law EXACTLY, but I've seldom heard of anyone being accused of poaching squirrels.
Now, that said, keep in mind that most municipalities have rules against the discharge of BB guns within the city limits, as well as it's very common that there be civil code prohibiting hunting within city limits. The only time you'd get in trouble is 1) if you got caught, basically because the neighbors complained, or 2) if you damaged someone elses property, which results in getting caught. Additionally, even if there is NOT municipal code (or otherwise) that explicitly prohibits BB guns or hunting, there may be other "catch-all" codes or laws that could land you in hot water, wreckless endangerment, public nuissance, negligent behavior, wreckless discharge of a weapon, etc etc.
On the other hand, if you're out in the country and want to blast a few squirrels with your Daisy, knock yourself out. I HIGHLY recommend safety glasses for any time you use a BB gun, as bounce back is a very common occurence... You'll put your eye out kid!
And all that said, a BB gun is a lucky shot to kill a rabbit unless it's a fairly high powered model. Squirrels and birds are about the limit for what you can effectively hunt with a BB gun. Pellets will increase your capabilities, but they're still not very powerful weapons. Beyond that, personally, small birds are nasty, filthy little animals, and squirrels and rabbits don't offer much meat. Sure, it's incredibly sporting and great practice for larger game to stalk and be able to connect on small game, but as far as a real food source, it's a lot of work for very little reward. Scientifically, I'd bet that hunting, cleaning, and cooking a squirrel requires more energy than the calorie content of the meat you get off of one!
But again, I'm all for the "boys will be boys" mantra, regardless of age or sex when it comes to BB gun hunting of rabbits, pigeons, starlings, squirrels, pack rats, etc...