Please help. Is the stray spreading something to my other cat?

I've recently taken in a stray. I've had her for about 2 and a half weeks. The other day I noticed she had yellow mucus snot type stuff that will come out of her nose or create a huge booger on her nose. I've only noticed this only one other time since I first seen it. I can easily get it off and afterward I'll check her nose and it looks clear, like a normal cat nose. She doesn't have a problem breathing or anything. She is very playful and doesn't seem to be unhealthy other than that. I've checked her every day at least twice a day. I've only seen this twice. Well, earlier today I noticed a bunch of thick white gunk in my boy cat's eye. This really bothers me because I don't want to see my babies sick and it's causing him to not even want to open it all the way. I got the gunk out and he seemed fine but after further inspections not only did it come back not even within a few hours but it was turning into a yellow mucus as well. His eye is watering on top of the mucus and he doesn't even want to open it all the way and it just really upsets me. My cat is an indoor cat and has had no problems prior to us bringing in the stray.  Does anyone know what this could be? Is there anything I can do to offer them relief? Is there anything over the counter I can get them or does this seem like something a vet should see? I'm really hoping to help them in any way I can. Thanks for any responses! 

Thank you for taking in this stray.
The rule of thumb is, if possible/practical, you should separate the new cat from the resident cats for at least several days. The stray may be purfectly healthy when caught, but because of stress, develop, for example, a URI. Also, depending on how long the stray cat lived on the streets, him/her having Worms is almost a certainty. As is pregnancy if female.
Also, the newbie may have something like FeLK or FIV, which can only be tested for at a vet's office (called a SNAP test).
You should take the new cat to the vets asap. The new cat could have an upper respiratory illness, which is high contagious.
It's also possible her has feline herpes. Neither URIs or Feline Herpes are fatal or anything--if treated sooner rather than later.
There isn't anything OTC that you should give a cat, certainly nothing like Benadryl. L-lysine is appropriate--but first a visit to the vets.
Keep us updated, and again thank you for rescuing this stray cat.

Sorry this took me so long to get back to. I have taken the cats in and the vet said that she had an upper respiratory infection which had spread to my boy cat but I got them all on antibiotics and they are doing well now. She tested negative for FeLK. All are doing well now. Thanks for the advice!

DON'TBASTRANGER!