Will majoring in Computer Engineering rather than Computer Science or Game Programming hurt my chances on getting a (game) programming job?

Your major will not matter nearly as much as your skill in software design and development. To the extent that your particular college's computer engineering major teaches you less about these things, or gives you less practical experience, that may hurt your chances; but you have to evaluate that WRT your particular program of study.

A college degree doesn't mean that you are going to get a job. It's only a stepping stone. College won't teach you everyone but will provide a base to work from. Most everyone I know who is in the programming field in some sort or other, learned most of what they know from actually working at different companies. In that respect you may want to look at where the upper level classes in that major will take you. At my school most of the upper level classes in the Computer Engineering field don't really correlate directly with software techniques. This I believe is a problem at my school, as right now their computer engineering course is nothing more than an Electrical Engineering course with a computer science minor.

I was struggling with this same topic, but I chose instead to go for a Computer Science degree and a math minor. Focusing on computer graphics and the neccessary math classes to complement that (linear algebra, calc c etc..)