I am no expert, but I would give him LOTS of reassurance and physical contact (as an aside do you have a pet? helping stroke an animal can calm people down massively, and will help hi realise pets get ill/die and that life goes on afterwards). I would also reassure him verbally, but not pandering to him. So he says "my brother has a red arm" you ask him why thats a big deal? he tells you he thinks he's going to die or something which is bothering him. I would laugh at his concern, make light of it, say DIE?! why on earth would he DIE from an injection. Not to ridicule him or make him feel weird, but for him to realise he is being silly, and needs to sort things out in his mind. I would then talk him through it using examples ie aunty betty had a hideous stomach ache, she couldnt breathe, was RUSHED to hospital and the doctors took out her appendix. 2 days later she was home, perfectly fine. So even REALLY SERIOUS pains can be dealt with, and look at aunty betty now, you'd never know she'd been ill - lots and LOTs of conversations like that, to show him that this is part of life.
I sympathise, because I am a fairly anxious person and do have hang-ups about serious illnesses and death, and I take on other people's worries. For me this has got worse as I have got older BUT, and this is a big BUT, I have managed to suport my mum throughmy dad's recent op for lung cancer, and support my dad through chemo, all of this happening just as I gave birth to my first dd and had the baby blues. I am able to "normalise" things and realise EVERYONE worries about death, ding, getting ill, but that these things I cannot control. So it's ok to worry about them from time to time, but not to obsess or panic.
I feel things deeply but shut down emotionally very quickly when having to deal with things that concern me. I am not particularly emtionally "intelligent", have a google and see what I mean about this, it is hard to explain.
But as I say, I am no expert, but having lived through this anxious life (without ever being medicated or seeing a professional btw, and not feeling the need to even mention this to anyone except you!) I think it is important that he realises this is not "normal" behaviour and learns how to rationalise his fears in his mind so he can relax.
Finally (i promise ) does he do much sport? sport/running helps me work off my anxiety and stress so I am more able to cope.