My take is that "You can't prove a negative" is obviously false, because any statement can be made a negative. It also conflates "Logically impossible" with "Possible, but utterly impractical."Proving negatives is an important part of formal logic and mathematics.
However, it does raise the point that some statements, such as "there is no teapot beyond the orbit of Saturn" cannot be proven, because it is impractical to do so. This is an important point with real implications.