As a VA doc, i can tell you the VA does a lot of screening for PTSD, TBI, etc and that is where those questions come from. The VA does not release any records to DHS, or local LE. That being said the VA has an excellent electronic medical records allowing a doc to access records from all over the VA system. Once something gets documented in the record its there forever. Could it be accessed in the future, sure. Would it be legal, no, at least not now. You should always be cautious about something that will last forever! And please don't answer yes to questions just to try to get some VA compensation.
I have never been asked any questions about guns in the home by my personal physician. I am always stressed when I go in for my yearly physical. Prostate exams are not my favorite thing.
I think most of the Internets tubes posts like the conspiracy theory that Clinton and the UN are going to take away our guns are scare tactics to get people to donate money to their "cause".
I can't comment on the VA because I am not a veteran. However, there is some reason to worry about information you reveal to your doctor. In some states, information you give to your doctor is not confidential. Some states have laws that require doctors to divulge certain personal information such as an admission of being an alcoholic.
I believe in PA there is such a law and a man actually lost his license because he told his doctor he drank a six pack of beer a night.
I found a link. This is real and is preposterous. This is like punishing someone for a crime they might commit.