What virus does my cat have?

i have a really kewl cat named days. he mainly lives outside and comes in when he wants to eat and sleep. then he goes back outside when he feels the need. i try to give him the three way vaccine every year but cant remember the last time i gave it. it could be well over a year ago. anyway about three weeks ago i noticed he had some green discharge in his eyes. bad enough that he kept squinting. so i treated that with an eye ointment and the drainage has gone down quite a bit. now its just minimal. but yesterday and today i noticed that after petting him real hard he'll crouch down in a position that looks like he's dry heaving but he's not dry heaving, he's coughing like ten times in a row. i tried to listen to his lungs and tracheal area with a stethoscope and they seem to be clear. i hear him sneeze sumtimes. i have the three way vaccine ready for him but dont want to give it til this clears up. have any idea what it might be? i have some type of nutritional supplement from the vet called enisyl-f for my indoor kitty that has nasal drainage. its suppose to boost the immune system. ever hear of it? think this will help?