Can taking cold eeze every few hours starting after you've been around someone who has a cold prevent you from getting sick?

Not really: Taking Cold-Eeze or other zinc lozenges won't prevent a person from getting sick from cold viruses. There's a lot of advertising for such over-the-counter products, with promising claims, but they don't really work. After being exposed to another person with a cold, one could get sick (if unlucky) or not get sick (if lucky)... regardless of whether one is taking zinc lozenges or not.