The negative claim that so-and-so has never eaten a baby is difficult to prove because that claim is actually a group of claims: so-and-so was not eating babies at time one, time two, time three, etc. Each and every claim in that sequence could be proven with a photograph that shows so-and-so doing something other than eating babies. Each negative in that sequence is provable. The problem is the length of the sequence: you need hundreds and hundreds of proofs. You would have the same trouble with any similar sequence of positive claims, such as the sequence entailed by the statement that there is a human-habitable planet in one out of ten solar systems.
That's equally difficult whether you're trying to prove that "All swans are white" or "No swans are black": the only ways to conclusively prove either involve examining large numbers of objects.
Go read Karl Popper on falsifiability.
I can prove there is a mouse that is blue and black on my desk, but I cannot prove that there has never been a red and black mouse on my desk, I can only assert such.
"You eat babies in your spare time!"
"No, no I don't."
The 1st can produce a picture of the person eating a baby, how does the 2nd prove that he doesn't eat babies?
If the person who had the claim made against them had to definitively prove that beyond a doubt he doesn't eat babies - he'd have to have had someone recording him for every minute of his waking life, including those moments prior to the claim.
The only way to prove a negative is to run a (perhaps nearly) infinite number of tests and never get that result.