Don - well I see Matt said it's not a big deal if the button breaks off when you attempt to remove it. He mentions you can just make up a new rod with a bend on one end and threads on the other. Well, to me that's a PIA. And besides that you will very likely need a new button as well, and most likely a new spring which is causing you problems now, BUT THE REAL PROBLEM is very likely the little washer that sits inside the tube below the spring. Its likely rotted away and allowing the spring to compress and stick so it doesn't push the button back up.
Have a look at the parts explosion on the Cub Cadet website so you can see how everything fits together. Here's a link:
https://www.cubcadet.com/webapp/wcs/stores/servlet/ARIPartFinderView?storeId=10051&catalogId=14101&langId=-1#/Cub_Cadet/122/Implement_Handle_%26_Connections/001860-A/0018600023
Your efforts to break down the rust using liquid wrench or any of the other concoctions mentioned on here are very likely not working since the button screws onto the top of the rod and covers the threads. To get the fluid into the threads you'll have to remove the lift arm assembly, plug the area around the button, turn the assembly upside down and pour in your rust buster fluid, and then leave it a couple days and hope the fluid doesn't leak out all over the place.
(Ooh, and getting the lift arm off is a BIG PIA since you have to remove the foot rests and that bolt that holds the front of the foot rest is a bear to get to on one side).
I see Charlie has the new buttons and new springs but doesn't list the washer which is NLA. You can probably figure out a washer that will fit and work. It should be just under the diameter of the tube hole, and have a hole that fits over the rod diameter. Once the button and spring is removed it slips in from the top of the tube and will drop to the point of the indentations (which might be covered up by the grip).
Hope this helps. Overall it's a great design, that IH toughened up as things went on, but it does succumb to rust over time.