Does continuing to take allergy medicine (allergy pill and nasal spray) prevent the flu or common cold?

No: Allergy medications have no role in preventing or treating flu or cold.

Absolutely not: Allergy medicine is great for treating allergies and the nasal steroids such as Flonase may provide some slight symptomatic relief during a nose cold. Allergy meds (or any of several other treatments advertised as preventing a cold) really do little to prevent the cold. The best prevention is the simple old-fashioned stuff, wash hands frequently, avoid sick people and rest and take care when sick.