Hi and thank you for the question today. I am a Registered Nurse and will be happy to help.
It sounds as though he is very sick if he is throwing up phlegm and all the other symptoms. Influenza can certainly cause some very unpleasant symptoms if that is what he has. There are also many other viruses going around in schools and neighborhoods that if not influenza, it could be another upper respiratory infection.
The blood red eyes could be secondary from not sleeping well, fever or dehydration. Or he could have a primary eye infection. Dizziness might also be a sign of dehydration. Pneumonia can be a problem if the infection goes on too long.
Sounds like the poor guy needs to be seen so that he won't miss any more school and he can start feeling better (and hopefully you won't get it either).