Currently, what is considered a "payroll tax" is not income taxes set aside, but a specific set of taxes called FICA taxes. This includes 6.2% of wages, payed by the employee, into the Social Security fund, and 1.45% of wages to fund the Medicare program - the nationalized single-payer plan for those of retirement age (generally). The total is 7.65% paid by employee. In addition to that, employers/companies must match that payroll tax withholding.
Obviously, since the current rate is 15.3%, total, combined, they're probably talking about raising it, probably by 7.35% for employees with a corresponding match (if we're being very optimistic and thinking it's the "up to" scenario). So a person making $80,000 would see a bump of $5,880 with the company matching, for $11,760 total shared contribution increase for my imaginary worker in this scenario. While the total for the California worker would go up to 15%, total, that 7.65% still goes into federal SS and Medicare, so that increase amount would be a one-to-one match on the funding pool for a California non-Medicare single-payer plan.
If employer and employee would both see their health care premiums completely disappear, that would be in line with the cost of premiums ($5K - $7K single, $13K or more for many family plans). Whether it would completely be a wash, or maybe a savings for some, would probably vary pretty widely, as did the benefits vs cost burdens of ACA. With deductibles and co-pays being used so aggressively to try and shift more costs from premiums to end-user (a shift from employer to employee, generally, because of the unequal premium split noted below), that adds another level of variability to any insurance plan. This is why legislation dealing with this tends to be so massive and voluminous.
Zane Blog:ACA Policy cost averages for 2017
So, basically, any increased costs would not fall exclusively on the employee, but certainly it would disproportionately, because in most markets, and especially with the job market skewing more towards the worker in recent years, companies generally pay more than a 50/50 split of health insurance premiums, so the offsetting premium savings would go to the employer.
It's my understanding that they don't have the specific funding mechanism detailed in the measures that have passed, so my speculation about the benefits and shares is largely hypothetical.