Legal Question in Employment Law in California

In most circumstances, yes. If they are paying 100% of the premiums, they can definitely require it. They can also require it if necessary to their insurance company's participation rules. Since I don't know of any other reason an employer would require it, in pretty much all cases the answer will, therefore, be yes.

Yes, if it is required under their coverage policy. Many are that way. In future, unless 'Obamacare' is repealed, you won't have a choice.