Broadly speaking, as summarized by Josh Barro here, there are two separate elements of the conservative take on Obamacare and the CBO’s findings. The first is that it allows people to break “job lock” and leave the labor market. This means there are fewer people working, which concerns conservatives because, as Ross Douthat put it, paid wage labor is “essential to dignity, mobility and social equality,” and they “see its decline as something to be fiercely resisted.” 
The second is that, because of the subsidies that are given to low-wage workers, these workers face a higher marginal tax rate. If there are subsidies for low-wage workers, as those workers make more money those subsidies are phased out. The fact that they are losing money while earning more money, or that a higher income means a smaller subsidy, functions like a tax. And this means that workers will work a bit less. Liberals in general don’t like this (though they do like that both effects will increase wages, as well they should), but understand it is going to be part of any type of means-tested income support.
Where does the minimum wage come in?
To address the first complaint, it’s important to keep in mind that the “dignity of work” isn’t a static concept, but tied directly to the conditions of work itself. If you ask the people striking against their low-wage job right now, you’ll find that things like working unpaid hours or erratic scheduling are also part of their complaints. As a result of these conditions, the work is socially tagged as undignified, degrading, erratic, and unpredictable. 
So driving the wages straight up can help counteract this. As Ron Unz writes, “consider the impact of a sharp rise in the minimum wage, sufficient to remove the taint of poverty overhanging so many of our lower-tier jobs.” This would, in turn, make lower-tier service jobs more attractive from a social perspective, increasing the level of dignity for those who hold them. This would in turn make people much more likely to seek out and hold said jobs. As I’ve argued elsewhere, by reducing vacancies, encouraging job searches and tightening the low-wage labor market, a higher minimum wage would also de facto give low-wage workers more power in the workplace, which would help reduce the petty tyrannies that come with low-wage work.