Technically, in the world of mathematics - the answer to this question is yes. There are negative factors and multiples of numbers. For example, -4 and -3 are factors of 12, -24 is a multiple of 3, and the factors of 4 would be -4, -2, -1, 1, 2, 4.
But what exactly does this mean in relation to the GRE and/or the GMAT?
For the GRE: This concept is mentioned in the ETS GRE Math Conventions PDF. You can find it here. The factors of an integer include both the positive and negative integers. This also applies for multiples.
Therefore, technically, if a GRE question were to ask "How many factors does 6 have?" The answer would be 8: (-6, -3, -2, -1, 1, 2, 3, 6). With that said, questions are very likely to specify "positive factors" or "positive integers" on the GRE. We haven't seen an official question before that required knowing that negative factors/multiples exist - but that doesn't mean those questions won't show up at all so you still want to be diligent about this. :)
For the GMAT: The same rules in the GRE apply to the GMAT in this case. Once again, the factors and multiples of an integer include both the positive and negative integers/multiples.
Once again, we haven't seen an official question that involved negative factors or multiples, but it doesn't mean that they don't exist. However, like the GRE, the questions will also clearly specify "positive factors" or "positive integers". :D