Employee pay would not go up much (or at all) if single payer healthcare becomes reality.
Why would it?
People, in general, like the status quo and are satisfied with it. As long as their pay doesn't go DOWN, they won't complain much (aside from a small percentage who realize how screwed they're getting in this situation.)
I'll give you an example: Auto insurance.
If you sign up for auto insurance with a new or newish car, you will get a competitive rate.
Then 5 years pass and your driving record stays clean. Your car is now worth significantly less, thus insuring it should be cheaper. Do your insurance rates go down?
Of course not. They either stay the same or go up.
Why? Because insurance companies know that if you were okay paying last year's rate, you'll be fine paying the same rate this year.
I used to switch auto insurance companies back and forth every 2 years or so in order to get better rates.