Yes, you can say "I am a boy." in several different ways in fact. As Kendra pointed out, anime characters say "僕（ぼく）" a lot, which is correct, but is usually used with informal speech. Depending on who you're talking to and your situation, the sentence can look very different. A couple of examples:
私は男の子です。（わたしはおとこのこです。） - from my understanding, this is the most correct way to say this sentence in です form.
僕・俺は男の子だ。（ぼく・おれはおとこのこだ。） - this is what you'll probably hear in anime. In this particular case, though, 俺（おれ） would be a very weird sentence, not because it's not correct, but because the people who use 俺 are usually older men, not boys. 僕 is usually used by younger men and boys alike.
You can also do this if you're really into talking too politely: 私は男の子でございます（わたくしはおとこのこでございます。） This, at least to me, sounds incredibly forced, but there might be differing opinions on the matter. Don't use this unless you have a situation in which you have to speak with some sort of CEO or something.
Hope I'm helpful, and have fun studying Japanese!