How to handle negative integer values in a data set?
There are no problem with negative inputs values, as long as it's mean something. As Ankit Saith said in the comment, temperatures can be negative, money too (positive is money I earn, negative the money I lose) and so on. Of course, inputs like distances should not be negative !

Furthermore, generally in deep learning, you normalize your dataset to have inputs with 0 mean and a std of 1. Then you have "small" value around 0 which can be positive or negative !