The thing to remember is that no living system is 100% efficient. Firstly lets deal with your statement here:
A positive nitrogen balance (where nitrogen balance = N intake − N loss) indicates a greater usage of proteins.
Basically means what it says on the tin. If your body is utilising more nitrogen then it is giving out:
Positive nitrogen balance is associated with periods of growth, hypothyroidism, tissue repair, and pregnancy. This means that the intake of nitrogen into the body is greater than the loss of nitrogen from the body, so there is an increase in the total body pool of protein.
Now we really cant measure notrogen on a weighing scale. So the blood urea level is used as an estimator for nitrogen metabolism.
Now, as mentioned below, no system is 100% efficient, and the purpose of urea is infact to be as a vehicle to excrete toxic ammonia. Sure the body looses a few nitrogens in the form of urea, but as a whole, it gains a lot more from the metabolism.
In other words, think of it like this: When you make something out of wood (proteins), you get wood shavings (ammonia). Now you cant reconstitute the wood shavings back to a full wood (proteins), nor can you use wood shavings (ammonia) as they are toxic for you. So you group up the wood shavings (urea) and get rid of it.
The formula is essentially:
The number of furniture made = Function of (Weight of wood delivered - weight of saw dust taken out)
If you use a lot of wood (proteins), and are a good craftsman (healthy human) you'll ultimately have a bit more shavings (blood urea nitrogen), but a lot more furniture to move around. Now since you cant see the furniture (proteins in tissues), but you can see the shavings coming out (blood urea nitrogen) there will be a rise in the BUN.
Now if you are a poor carpenter (disease or something), you mess up and you end up wasting a lot more wood as shavings than getting the furniture right. So for say 10 kilos of wood, you can only make one couch and 9 kilos of saw dust.. you are really not doing it right, and this is where the BUN gets in the negative territory (I know 10-9 is not negative, but remember the body has a baseline need of say 6 couches and you are only giving it 1, the body will get that from somewhere to the detriment if your health)
Apologies for the long winded analogy. I hope you understood what I was trying to imply. I am pretty sure someone out there will give a more scientifically accurate answer, so you might want to wait on that also.
Further reading from the wiki:
Amino acids from ingested food that are not used for the synthesis of proteins and other biological substances — or produced from catabolism of muscle protein — are oxidized by the body, yielding urea and carbon dioxide, as an alternative source of energy.
The oxidation pathway starts with the removal of the amino group by a transaminase; the amino group is then fed into the urea cycle. The first step in the conversion of amino acids from protein into metabolic waste in the liver is removal of the alpha-amino nitrogen, which results in ammonia. Because ammonia is toxic, it is excreted immediately by fish, converted into uric acid by birds, and converted into urea by mammals.
Ammonia (NH3) is a common byproduct of the metabolism of nitrogenous compounds. Ammonia is smaller, more volatile and more mobile than urea. If allowed to accumulate, ammonia would raise the pH in cells to toxic levels. Therefore many organisms convert ammonia to urea, even though this synthesis has a net energy cost. Being practically neutral and highly soluble in water, urea is a safe vehicle for the body to transport and excrete excess nitrogen.