In thinking about the effects of ACA on employment, I have neglected the effect of the Medicaid expansion.
Overall there are a multitude of effects of ACA on the supply and demand for labor (that framework of course being my model of choice for the analysis). On the demand side, we should expect that the employer mandate in the ACA -- provide acceptable insurance or pay a fee -- should decrease the demand for labor. Offsetting this negative effect on demand is an increase in the supply of labor, reflecting the value that employees get from the mandated coverage. As a first approximation, these two effects would perfectly cancel out, causing the money wage to fall and employment overall stay constant, but that is only a first approximation. It should be expected that the value of the insurance to the employee is less than the cost to the employer, as many employers are observed to not offer insurance. Also, there is the small problem that with very low wage employees, the money wage cannot fall so with demand being the binding constraint employment will fall.
A confounding effect here is caused by the subsidies offered to employees who buy insurance on state exchanges -- in some cases, the subsidy will be worth more to the employee than the fine that the employer will have to pay if their employees avail themselves of a subsidy (there are tax effects here too that I will ignore for now).
These are the effects that most analysts seem to take into account when looking at the effects of ACA on employer-sponsored insurance, see for this Urban Institute article or this article by Holz-Eakin and Smith.
But for the effect of ACA on employment, one more variable is important, and that is the expansion of Medicaid coverage. Right now, of course, Medicaid coverage in many states is quite meager, with able-bodied adults often if not typically ineligible. With the new Medicaid coverage, adults will be covered by Medicaid if income is below 133% of the Federal poverty level. In comparing the decision to be unemployed pre-ACA, the cost of health insurance (or the cost of not having it) would have loomed large. Post-ACA, the difference between the cost of insurance if unemployed is zero (Medicaid coverage) as it is if employed (subsidized or employer sponsored). This seems like it would be a rather large impact on the supply of labor -- reducing it, as one of the major costs of being unemployed has fallen dramatically.