There's a tremendous difference between what government should be allowed to do and what individuals should be allowed to do. O'Reilly is attempting to blur the distinction, a common rhetorical tactic but one which does not advance his argument. As far as I can tell, his only argument besides this is that if the citizenry pushes for the government to use Free software, companies will push back to use proprietary crud. This argument doesn't hold water - every company selling proprietary software is lobbying the government all the time, have been for years, and they aren't going to stop just because we do. CNet carries news today that Microsoft has pressured the NSA to drop development of Security-Enhanced Linux. I can only imagine what sort of pressures might have been brought to bear behind the scenes, perhaps Microsoft threatened to cancel the NSA's site licenses of Windows and Microsoft Office. But in any case, there's no such thing as "mutual disarmament" - if we back down we'll just get smashed by the continuing efforts of companies pushing proprietary software.
But back to the government/individual distinction. Individuals, for instance, shouldn't be required to disclose their private papers to anyone who asks. But government should: that's the foundation of our freedom of information laws, and they exist for a good reason - keeping an eye on government is a necessary thing. Saying "People should be free to keep their papers private" as an argument against government FOI laws is just a stupid strawman, unworthy of further debate. And that's what O'Reilly's argument against California's proposed law is as well.
Governments play by different rules. They need to be fiscally responsible, transparent to the public, and promote the public commonwealth whenever possible. Using Open Source or Free Software in government promotes all three of these goals, and if Microsoft or any other corporation doesn't make quite as much money when the government alters its standards for software procurement... so what? Companies who make shoddy products do lose business when the government ups its standards, and they have the same choice as any business does: either produce better products, or lose the government's business. In this case the shoddiness comes in some of the most important areas as far as software goes: open access to the code, to ensure the software that we the citizenry pay for is doing what it is supposed to be doing, but the rationale would be the same if the government mandated a certain level of bug-free-ness or a certain level of performance for software - you can shape up and continue selling to the government or you can ship out. Your choice.
O'Reilly seems to be promoting the agenda of Microsoft's Software Choice campaign. He's a business man; perhaps there's a reason we don't know about. But whatever his motives, his lame arguments are no reason to stop pushing for governments to use Free or Open Source software wherever possible.