My cooktop has white, black, red & green wires. The 40 amp outlet has black, white & bare ground wires. What connects to what?

You cannot use this circuit for that cooktop.
The circuit is a straight 240V circuit. The cooktop needs a 120/240V circuit. You do not have enough wires to do what you need.
And before you possibly say it, even if the old one was wired and worked either means it was a 240V cooktop, or it was wired wrong.

Your new cooktop requires a four wire connection. What that means is a new branch circuit 3/w ground cable (#8/3 AWG with ground) needs to be pulled from the panelboard's breaker to the outlet box.

The breaker does not need to be replaced. However, now four wires are needed. Black and Red serve as Hot conductor, White is the Neutral conductor and the Green or Bare conductor serves as the Equipment Grounding conductor.

Older circuits still required a neutral, but the ground was allowed to be left out. The neutral served both purposes.
A bare ground was never allowed to be a current carrying neutral.

The wiring configuration she is giving is a violation unless she has two black conductors. She states that there is Black - White & Bare in order for that to be a 240 circuit, both the Black and White conductors have to be ungrounded.

Of course it is possible that the original cooktop was a straight 240v appliance which would not have required a neutral, correct?