I had a cracked plastic push screw on my JT and replaced it today finding it relatively easy using bits and pieces from the junk drawers. If you've ever done this then the photos will probably makes sense. If not, well you may find it helpful once you have parts in hand. I used a good part from my other JT to ensure I put it back together correctly.
I basically made a puller or separator tool using a long screw that matched the threads inside the push screw boss. Once assembled I simply tightened the tool and the swaged end of the push screw boss pulled out of the pivot arm nice and clean. I used a spark plug socket that was large enough to swallow the plastic screw and black rubber seal. I modified a large washer making it a U shape with an ID matching the black rubber diameter. I slipped that washer around the the black rubber and as you tighten the tool it seats against the pivot arm and allows the force to separate the arm from the boss. I worked great.

Once I had the push screw off the pivot arm I then made a similar tool for removing the black rubber ring that I wanted to use again with the new push screw. I found a washer that had an ID just large enough for the push screw boss to fit thru but not large enough for the black rubber ring. I used the same screw from the other tool and found a socket that was large enough to swallow the push screw boss. I used a few properly sized washers behind the socket. As you tighten the screw the tool pulls the black rubber ring and the plastic screw off the boss nice and clean.

Now it was time to put the new push screw back together with the black rubber ring and attach it to the pivot arm. Again I used the same screw and some appropriate size washers along with an automotive brake caliper sleeve as a tubing spacer. I used a small washer under the white plastic screw making sure it was small enough to only seat on the metal boss and NOT on the plastic screw. Then I tightened it all up and clamped it in the vice with the jaws only touching the caliper sleeve, staying clear of the plastic screw. I put a steel block under the screw head to withstand the hammer blows. I inserted the steel ball back inside the screw boss and peened the push screw back onto the pivot arm. I had to treat myself to a new ball peen hammer that was small enough to peen the outer edges of the screw boss. Make sure you have the offset arm and the screw spirals in the correct position. I had a good part from my other JT to use as a guide. Taking photos before disassembly would work as well.

Thank you very much for a wonderfully clear method of repair with no special tools!
I was completely stumped as to how to go about this.

One matter: As I understand it and as per yamaha parts diagrams the steel ball goes onto the other end of the push rod not into the push screw.
Any comments?
Thanks
Roger

I have just realised I am talking about a AT1 125 which has similar setup to your JT but perhaps the location of the ball is different and that probably explains the comment.

Also on the AT1 it appears the lever/push screw peening exercise has to be done with the assembly in place in the side cover as one cant fully unscrew the push screw/lever assembly once in because the side cover has an internal web that gets in the way.
Poor design IMHO unless I am misunderstanding something....

All sorted. (previous owner had a makeshift arrangement)

By the way its metal push screw upon metal side cover. No plastic involved.