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Rawls observed that pretty much everyone has a sense of self-interest. No matter who you are or what you do, it is natural to want to help yourself. This makes producing justice a bit difficult - if everyone inherently skews the system in their own favour, unless all the skews balance out you're just left with a completely warped system.

What a rational system of justice requires, then, is some kind of barrier to protect the system from the above kinds of favouritism. It needs to be the kind of system designed by people who didn't know it might apply to them, or who didn't know how they would fit into the system. This is his 'veil of ignorance' - it's a kind of thinking which accommodates difference without your specific personal interests entering.

I suppose it could an argument against utilitarianism in the sense that it doesn't really exist. It's a theoretical position... but unless you do have some dispassionate arbiter to administer utilitarianism according to whatever ethical calculus you are using, it can very easily break down and become corrupt.

I suppose the best counterargument is that this isn't a criticism of utilitarianism but of ANY form of government and justice. Being a democracy or a monarchy (for example) won't eliminate the possibility of self-interested abuses of the system. In fact this is one of the more common complaints in the practice of either system - that a majority or the king is cramming their own personal interests down the throat of everyone else instead of in a way that they wouldn't object to if positions were reversed.