I don't think I agree. Mars has a lot of things going for it that Earth doesn't.
For one their are no explosions, or people with guns. On mars you don't have to worry about people trying to steal your shit because it's valuable (or "necessary for the national defense"), and you don't have to worry about being collateral damage during the war.
On mars there aren't earthquakes, hurricanes, tornadoes, mudslides, tsunamis, floods, volcanoes, lightning strikes, forest fires, and basically any other natural disaster that I missed except sandstorms. When you build a static structure on mars you can expect it to last basically forever, when you build one on earth it has a relatively short halflife. When your static structures become extremely valuable and difficult to replace because you need them to grow your food and you don't have the manufacturing capability to build more, I'd rather be one mars.
On mars you can reliably avoid radiation by putting enough "stuff" (water, soil, whatever) between you and the sun. On earth after a nuclear war it would be coming from everywhere exposed to the atmosphere (in the shortish term).
On mars you get to plan for the hardships while the pre-nuclear-war earth manufacturing base is still functioning. On earth you don't know when the nuclear war is going to happen, or what portions of the earth it's going to glass, so that becomes much more difficult.
On the downside, the martian atmosphere is actually deadly to humans (since it's nearly a vacuum), while the post nuclear war atmosphere is likely only ... very hazardous. Mars has no native plants so you have to be entirely self sufficient inside your colonies, while the post-nuclear war earth colonies could forage for (mildly radioactive) food in emergencies.