Asthma is classed as a chronic respiratory disease. Allergies do affect asthma and can bring on asthma attacks or asthma related symptoms like coughing, tight-chest etc.
I havent heard of Zyrtec myself but if your son is on this medication for allregies then depending on his allregies and how they affect him, what times they affect him (eg all year, few weeks during summer etc), then this will boil down to how long he will need to be on Zyrtec.
Its possible your son may have asthma. Asthma varies from mild, severe and moderate. Allergies and whats the weather is like can affect asthma too. There is a test your son can have. Its called a allergy test. He will have tiny pinpricks on his skin and he will be watched to see if this causes redness, itching or any other symptoms. This allergy test is a MUST for Asthmatics. It really helps to let us know what we need to avoid whether its everyday dust, pets, hayfever etc.
I would take your son back to his doctor and have him explain how long he needs to be on the Zyrtec for. Also ask the doctor to test your son for Asthma. If he has got Asthma and allergies are causing it to flare up, the allergy test is advisable and the Asthma medication will help your son greatly. Take care!