To what degree should I expect advanced calculus to be used in mechanical engineering positions? Along those lines, which engineering specialities are the most math intensive?

Every path is a little different, so you are likely to see a range of answers. You'll find it is less about the engineering specialty and more about the type of work on which you'll embark. Work in a manufacturing environment, technical sales, or industrial design where safety factors mean failures are extremely rare, and you are likely to not find many using differential equations on a consistent basis. Modeling software is fairly common. Most people building consumer products with plastics are more likely to model and prototype than spend significant time developing unique math models. Its not just a valuable use of time on a highly driven delivery timeline.

While living in LA I had friends that worked in the space and aeronautics industry, and they would spend a lot of time designing a single part to be completely optimized to stress conditions with less care on the ultimate cost. Performance was king, and they would use models and mathematics. Often times, there are simulators that dramatically speed up the process so you may not personally do most of the math. You do need to understand how the models operate and know the principles to recognize issues. If you end up at a company that makes the modeling software, obviously your use of differential equations goes dramatically up.

Depending on your interest, you may find similar engineering fields more interesting including materials and aeronautics. It would likely be in your interest to evaluate the many engineering disciplines that support the space industry.

If you are highly interested in high performance systems where math is used to optimize design, you may consider contacting one of the several suppliers to NASA (Raytheon, Honeywell, Boeing,etc.) and talk to a university recruiter about which fields and universities they recruit from the most. Large companies usually have university preferences with sponsored partnerships. Engineering programs are specifically designed to support the partner industries with experts on the teach staff. You're more likely to see software experience, class projects, and internships that will land you in the job you want.