How do you convince students that it's OK to teach them "how to think?" Often in a Calculus or Linear Algebra class, my students may ask - when am I ever going to use this?  The answer I try to give them is "You're not. And that's totally OK. But you do need to learn how to think logically and solve problems. Mathematics provides a safe place to learn these thinking skills."

But increasingly, the focus of the students is still "Is this going to be on the test?" 

They seem unhappy when questions they haven't seen before end up on the test. My thought is that if they learn the process for solving problems, then they should know how to approach a new (but similar) problem. 

I'm not interested in seeing them regurgitate facts, but I want them to develop their thinking skills and use them to solve new problems.  This often causes backlash.  I get it - learning how to think is harder than memorizing facts.

How do I effectively communicate that it's not the individual questions or ideas that are important, but it's the learning and thinking processes that are key?
I disagree with your fundamental premise: that the concepts won't be used, and therefore the main benefit of learning calculus is to hone a student's problem-solving skills. 

Many engineering problems do require an understanding of calculus. Perhaps not all your students are destined to be engineers or physicists, but, if you explain that there are people who do need and use this knowledge to advance in their career field and solve real-world problems, you might get them to sit up and realize they aren't trudging through their homework as a mere academic exercise. Fact is, some of these courses are taught to freshmen and sophomores because they will need those problem-solving skills as juniors and seniors – not just to know how to solve a problem, but to also understand the underlying theory and mathematics. If all the students in a particular demographic will never use calculus in the future, then they probably don't need to take calculus now. 

I'd try to find some real-world problems that require the knowledge taught in your course to solve. Perhaps that might spark an interest and help students realize that the are not just solving integrals for integrals' sake. 

If I was at your university, and I taught an upper-level engineering course in heat transfer or thermodynamics, I'd be disappointed to learn that my students' calculus teacher thought the main reason to teach my engineers calculus was to provide a "safe place" for them to "learn how to think logically and solve problems."