Installing timer switch with jumper wire into possibly old wiring
You got the wrong switch. 

A spring wound timer switch, like the GE unit you're taking out, is you-powered.  It doesn't need energy from another source to do what it does. Like any passive switch, it is simply a gate between "always-hot" and "switched-hot". 

However, your push button timer switch needs power to do its little thing.  And it needs power at all times. In electricity, you get power between hot and neutral. 

So the switch needs neutral.  Your switch is on a switch loop, and historically they never provided neutral on switch loops. Now they're starting to do it... but if you don't have neutral, you can't use this switch.