I was trying to play devil's advocate but I honestly couldn't think of a good reason to teach Calculus using epsilon-delta vs. Nonstandard Analysis. The way Calculus is taught today is already unrigorous until you get to Analysis, so there's no strong reason to teach students using epsilon-delta first.
Right, because students struggling with "for all P there exists Q" won't have any problem with higher-order logic?
It's always seemed to me a more sensible road to both simpler and more rigorous calculus courses would somehow involve reducing the scope of functions under consideration, since most of the exercises involve analytical functions anyway. Then, when students are ready for analysis, it can be more about "how to reduce nasty cases to problems you know how to solve, and how to recognize the truly pathological specimens where you can't" rather than "everything you thought you knew is wrong."
You don't need higher order logic to present a nonstandard analysis approach at the same level as of rigour as a standard limits-based calculus course. Keisler wrote a great infinitesimal-based Calc book (http://www.math.wisc.edu/~keisler/calc.html).
In terms of pedagogy I've found that there's a huge leap that students make between Math focused on computations and anything involving proofs. The reason it's difficult to make Calculus rigorous is not because you have to address a ton of cases-it's because understanding proofs is really hard. This is why students who take Algebra and then Analysis or vice versa tend to do much better in their second course-because they're already used to proofs. So I don't really think it's possible to make a first-year Calculus course more rigorous by sticking to analytic functions, because you still have to get over the proof barrier.
