The problem with publicizing genetic information is that you're defacto publicizing large amounts of the genetic information of your relatives, who may not be in a life situation where publicizing it carries no risk. This is also an objection many have to 23andMe.
What if they live in a country in which genetic evidence of a disease can deny or significantly increase the cost of health coverage? Even if you're clear of those for now, a new marker may be discovered tomorrow. Apparently (according another commenter) Life Insurance /can/ legally look at this even in the US. What about employers? What if it puts them on the DNA-evidence hook for a "crime" in their jurisdiction which you and they don't think is an ethical law (evidence of homosexual activity in a country that imprisons for it, or worse).
The crime thing sounds like a huge stretch given it's not actually your DNA.
With the insurance example I'm not sure I have a problem with that? The whole pre-existing condition conversation around health insurance is totally out-of-whack. Insurance was not designed for things you know have happened. It was invented to reduce the downside of things that could happen, commensurate with the risk of that thing happening. It's risk management. It makes zero sense to apply that model to something like universal health coverage. If someone is 100x more likely to get cancer, their insurance premiums should be higher. Just like if I'm a ship captain sailing into the bermuda triangle my premiums should be higher than sailing around the mediterranean.
If you feel that everyone should have healthcare, utilizing health "insurance" for this is the worst of kludges.
For example?