It is good to dispel myths wherever you find them, and, unfortunately, this is one of them. You can certainly prove a negative. Take a box. Make the proposition "There are no unicorns in this box". Investigate the box. Find no unicorns. In fact, every time you prove a positive (A), you are actually proving a negative (~(~A)).
http://www.skeptic.com/eskeptic/07-12-05.html (see the article "You Can Prove a Negative").
Strictly logically speaking, you are always proving a negative in science.
A -> B
~B
Ergo, ~A
If rain (A), then wet sidewalk (B).
No wet sidewalk (~B).
No rain (~A).
You just proved a negative.
Now when it comes to
universal negatives, it gets a bit tricky. You can prove that no unicorns exists in the box, but can you prove that no unicorns exists at all?
"1. If unicorns had existed, then there is evidence in the fossil record.
2. There is no evidence of unicorns in the fossil record.
3. Therefore, unicorns never existed."
Provided the other person agrees with A and B, you have proved a universal negative.
When it comes to gods, one can certainly disprove some gods, like the young-earth creationist god. If you answers that god-is-a-trickster, one has suddenly made an ad hoc hypothesis and changed the definition of god. This new version is not disproven, although the old one seems to be.