Right now I can get a 1yr government bond that yields 0%. Is there any reason to buy such a bond? E.G. If the broker holding my accounts goes out of business is my "perceived low risk portion" of my portfolio any safer in a government 0% bond than it is in a money market account held by my broker?
0% bonds are desirable for some individuals. It depends on your situation. 0% bonds are usually sold well below par value (eg a 100$ face value bond for 2020 might sell for 90$ today) Hence, your gains will be CAPITAL GAINS. A similar investment paying interest would be taxed as INCOME, and smaller portion of capital gains. In many countries (US, Canada) Capital gains are taxed at a more favourable rate then income. This is especially true when holding these investments in corporations.
No, there isn't.
There are a number of reasons that institutions buy these bonds but as an individual you're likely better off in a low-yield cash account.
By contrast, there would be a reason to hold a low-yield (non-zero) bond rather than an alternative low-yield product.