The product of the (positive and negative) divisors of an integer $ a $ is negative if $ a $ has an odd number of negative divisors. It follows that $ -a $ must have an odd number of positive divisors. However, for every positive divisor $ d $ of $ -a $, then $ (-a)/d $ is also a positive divisor of $ -a $, so that the positive divisors of $ -a $ can be paired up. The exception is if $ -a $ is a perfect square, in which case $ \sqrt{-a} $ will not be paired up with another divisor. There are $ \boxed{14} $ perfect squares between $ 1 $ and $ 200 $: $ 1^2, 2^2, 3^2, \cdots, 14^2 = 196 $.