The Bible of the history of mathematics is Morris Kline's Mathematical Thought From Ancient To Modern Times,but this is really for mathematics students.I think it'd be hard to teach a course from it to non-mathematics majors.
To be honest,I think it'd be really hard to teach such a course to non-mathematics majors PERIOD:How would you explain why Wierstrauss' nondifferentiable function is important to sudents who don't know calculus?Even worse-why would such students CARE?
If I was really gung ho to teach such a course-I'd take Kline,Stillwell,Bell's Men Of Mathematics and 3 or 4 other texts and use them to write a set of lecture notes for my students.That's how I'D do it.