"For some it is uncomfortable to learn that, following the most recent developments, there are British nationals who previously would not bother to look for a different nationality but are now eager to."
It is not at all uncomfortable for me. It is a perfectly reasonable response from people who want to remain a citizen of an EU member nation (and OG correctly points out the significance of that vs "EU citizenship") post-Brexit. It is far more reasonable than expecting the UK to remain in the EU (or to agree, on leaving, to anything which smacks of EU membership in all but name) when the majority of those who voted said they wanted to leave.
"I know of someone (a born and bred Brit) who intends to apply for citizenship of a country which is not in the EU,..."
Er..why? He doesn't need to. He can stay in the UK and after next March he will have citizenship of a country that is not in the EU.
"I despair at some of the anti-EU stuff on here to be honest"
You may well despair but you should not be surprised.