General engineering vs. specified engineering - pros and cons of each?

While I think engineering is a great field to study (I did so myself) there are so many career options afterwards that it's hard to say that one limits you much more than another. It will really depend on what your career goals are and how engineering is going to play into that.

As far as I know, most of the schools with strong engineering programs focus on specific engineering disciplines, rather than a general engineering degree. The reason for this is that although there is a bit of overlap between different disciplines (Ex: Mechanical, Aeronautical, Materials all have structural engineering courses) the way the material is taught and the purpose of it will be different for each discipline (Ex: Car Engines vs Airplane Wings vs Microprocessors). With only 4-5 years to get a student up to speed, it's really important for the school to focus.

As far as career paths, some careers will require a degree in a specific field, while others appreciate the general analytical thinking that engineers brings and won't be as picky. This is why it's important to get a good idea of what you want to be doing afterwards to help guide your decision on your major. Don't worry if you think it might change, very few people stay in the same exact field they start out in, but it's good to get you going and focused on learning something while in school, instead of bouncing around from major to major and having a hard time finding a job afterwards.

Hope that helps!