Taking zinc all the time doesn't do anything good for your immune system. Whoever told you it did is either intentionally deceiving you or just isn't smart enough to interpret the data.
A sudden increase in zinc intake generates an enhanced immune response in the form of elevated production of neutrophils and killer Ts (two generic types of white blood cells). This is particularly useful in fighting diseases to which you have no pre-existing targeted immune response (zinc doesn't do anything for your production of antibodies). However, if you establish a new higher baseline of zinc consumption your body will, within a few weeks, go back to its normal rate of white cell production. With this new baseline zinc consumption, you can't bump it up again without risking potential zinc toxicity. In effect, taking zinc every day only removes it as an actual useful tool to fight disease. The right way to use zinc is to start taking it whenever you feel like you might be getting sick.
All of what I just said is technically predicated on the presupposition that your normal zinc intake is normal/sufficient. If you are zinc deficient then taking a zinc supplement on a regular basis may in fact be necessary to maintain a normal, healthy immune system. But if you aren't zinc deficient, it's actually counterproductive.