Does the break even analysis contradict the law of supply? The law of supply shows a positive relationship between price and quantity 
However, a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even.
Does that mean that a firm producing at break even level is an exception to the law or that the analysis contradicts the law of supply ? ( please correct me if there are any inconsistencies in the logic)
The short answer is no. 

The law of supply only states that increases in price result in higher quantities supplied, ceteris paribus. Break even is the point where Revenue = Costs. 

Your confusion might come from this sentence:


  a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even.


If the price lowers, the firm will reduce its quantity supplied. That's it. The specific quantity depends on the variable and fixed costs. 
It's not mentioned that the company has to break even - think of a monopolist for example.