Lee.... I am a Licensed Pharmacy Tech in the state of Indiana 17yrs now. I can answer this for you on 2 levels.
A) Although I am not familiar with Piriton tablets here in IN, it looks to me, when I did a search, like the active ingredient is Chlorpheniramine..... So I would say NO. Do not give that to your jack.
B) I too am a jack russell owner. "Marshall" has horrible allergies and suffers from "hot spots" every spring, summer & fall. I give him childrens chewable Benadryl. the active ingredient is Diphenhydramine 12.5mg, every 4 hrs as needed. Hes not itchy! And wayyy more mellow :)