Is there a significance to a black panther tattoo on a National Guard soldier in from a 1960 photo?

It all depends. If it was a classic Black Panther tattoo where it is clawing up or down your arm, then it means nothing. If it was just the head of a black panther or a fairly cartoonish looking panther lunging at you with claws, then it very well may have been the symbol for the Black Panther Party. They organised and were created in 1966, so if it was before then, probably not. They did have a number of white people as well as others who were sympathetic to their plight and supported them, so the color of the person wearing it is fairly irrelevant.