If your son was on the antibiotic for 10 days before developing the rash, I highly doubt that it was an allergic reaction to the antibiotic.! Also, if the rash is still coming back when you discontinue the Zyrtec, the antibiotic that he took probably has nothing whatever to do with it. Any antibiotic that he took would definitely be out of his system by now!
I have dealt with terrible allergies my whole life, both in myself and with my son, who is now grown. My guess would be that your son is allergic to something else in your home, or that he's eating, and the only way to clear up the rash while stopping the Zyrtec is to find out exactly what it is. Is there anything else that is new in your home that would have been brought in, or occurred around the same time that he started with the rash?
Have you used any new laundry detergent on his clothes, or different cleaning products, or things like that? Have you gotten a new pet? Or did you start him on any sort of different foods around that time. Even using a different brand of food could be suspect, as preservatives added to the food may be different.
If you can't narrow anything down as a possible cause, then I would take him back to the allergist, and tell the doctor that you want allergy tests done to determine the cause.
While Zyrtec may be perfectly safe for him to take long-term, it is only masking the problem, not getting rid of it, and the longer he is exposed to the allergen, the more allergic to it he will become.
Once allergy testing has determined the cause, hopefully it will be something easy to keep him away from, such as a cleaning product, detergent, or something that he's eating. But he could also have developed an allergy to an airborne allergen, such as dust or mold, and if that turns out to be the case, then allergy shots on a weekly basis will build up his immunity to the allergen, and the rash should go away.
This is not something that you should just continue to give him medication for and wait it out. Not doing something now will just allow the allergy to become much worse!
I, myself, have an allergy to mold, which developed in my 20's, and before I began taking the injections, I had hives all over me the size of apples. I itched constantly, even taking medications. Today, after having the injections for a period of time, the allergy is controlled and seldom bothers me, so I would highly recommend this plan as something that you should undertake as soon as possible.