Medicare HMOs is another way to work with Medicare. There are various Medicare supplements, but many of them are not that good, and many have very limited prescription coverage. There are also some supplements available through an employer's group coverage, but not available to the public. Usually a very small amount. A way to get them coverage is to find an association that has policies that will coordinate with Medicare.
Many people read that and interpret it in a more convenient way, and then they get into trouble.
Yes, because their insurance company will deduct for that coverage, or cancel the coverage which they had previously. Yes, you could lose eligibility for your previous insurance. It depends on the individual policy. This doesn't mean that your options are closed. It means that you need to find a different way to continue to get full coverage.