Unless you need the inside mirror, in which case there are two different ones. The one used in the 1951-52 J, which did not have a dome light, mounts to the roof panel through the headliner. The type that was used on the 1953 Js is different. Because the '53 J has a dome light, it mounts on the center windsheild bar (which is also why the '53 four cylinder cars have a single chrome bar down the center of the windshield, they need it to mount the mirror). They are different - different stems for different mounts. I may have one for the '53 style, which could be used in a '51 if you don't mind drilling the center bar and / or leaving/patching a hole in the headliner.
But... if you need an outside mirror, it won't matter. I will say the hanging type as joefrazer mentioned would be best. Both of my Js have after-market door mounted mirror and you really can't adjust them to where they are useful. The clamp on type can if I remember correctly.
