Can you 'suggest' a mother take her son to a dr?

I'd have a chat with the teacher. They might be able to approach the family to offer ideas/ support and to work with the children on hand washing etc

Runny noses can simply be attributed to the cold weather. It has to do with the lining of the nasal cavity.
I can't remember it all exactly but we just learned about it in biology...
something do with the cellular and tissue struture has a hard tim working in cold weather - it's normal function is to hold nasal mucous where it belongs until you blow your nose, and explains why in cold weather we get the 'sniffles'.

FWIW i dont necessarily think this is what this kids problem is but working in childcare there are kids that look sick all the time and their parents take them to to the docs and its nothing - for reasons such as these. If you are really concerned i would be speaking to the kinder leader as well, and encourage the other parents that may be making remarks to do the same.

Id say that Id heard of the bad flu going around and that it was really nasty for some people, and the doc said everyone with a sniffle should get checked just in case

The green snot/contagious thing is definitely a myth. Viral infections rarely have green snot. It's usually clear. Bacterial infections (the ones doctors can help with) have colored snot.

I tend to have several months a year where I am a mucus production factory. It's not so bad over winter now that we are living in a different place but the last house we lived in the combination of allergies and the cold weather had my nose constantly on the go. DH has the sniffles every morning. So I don't tend to assume "sick" with mucus unless there are other signs.

I think if its just a gross nose, and not other signs the child is actually sick, I'd just mention it to the teacher as a hygiene thing and teach your own kids how to keep their nose clean. Not sure about yours, but mine is a wee bit shirt of tact, so where we've taught her about wiping her nose, if she sees another little person with a snotty face she tells them to get a tissue - maybe your boys will so the same!

I would say something. Only because I know of a little girl who had prolonged snottiness and it turned out to be a cancer in her nose. And you just never know.