Not typically, but it's possible that someone might be able to negotiate this with their bosses.

Typically people opt out of company insurance because of one of the following reasons: It's cheaper for them to find services through the marketplace (extremely uncommon, but possible), They have superior insurance through a spouse or family member, they feel that the insurance they're provided with isn't sufficient and they would prefer to purchase better insurance themselves.

No, I don't get the difference. The company keeps what it would pay towards my insurance but I don't have anything deducted from my check for what would be my share of the insurance either.

Edit: My wife though pays more for her company insurance as they would need to insure me as well.

It's not the norm, but I was once offered a job where if I declined health insurance they'd pay me the difference - it was a 4 person company.