Hi John,
You have an older generation transponder with the old solder on BNC (the later ones are simply a BNC to BNC which require no soldering).
That being said, there is normally no need to replace those connectors. You can desolder/remove the pigtail that is there and simply solder on a new one. Here's how to go about it:
1) That connector has a "cap" on it at the very back. It is held in place via friction and sometimes a dab of silver solder. If there is a dab of solder, remove it.
2) Snap off and remove that little cap (it looks like a plug that you use in electrical box cutouts, only smaller).
3) Now you can see the inside of the connector, where you can reach down with a narrow tup soldering iron/pencil and remove the coax.
4) Note that the shield will be soldered on the outside of the connector and also needs to be desoldered.
5) Once the old coax is removed, you can strip and new piece and resverse the process.
Note that if you don't have a "solder sucker" or solder wick, it's easiest to remove the inside center conductor first, then melt the shield solder and pull the coax out. Clean everything up good before starting to reinsert a new coax.
Anyway, if you buy a new one you'll still need to solder on a pigtail. This is not difficult, but also it is not trivial either...it needs to be done right so the pigtail is correction fastened and supported to keep it from breaking off (happens to many amateur installs).
If it were me, I'd just call me and order a new backplate with the new style BNC on it, then you can simply put a BNC on the antenna coax and push it onto the transponder connector like any other BNC connection. No soldering, no desoldering, no wondering if you got it right.
Lastly, you could just throw the backshell in the mail to us and we could replace it for you. Thing is the labor probably will end up costing you almost as much as just getting a brand new backshell....
There are your two options!
I know it's a short definition, but if you look closely at the manual it'll sort of be self explanatory as to how the antenna is soldered into the connector.
Cheers,
Stein