Can I use 2 CR2032 batteries to power an EL light strip normally powered by 2 AA batteries?
The first thing you need to realize is that your blue strands are EL strands, and that the green strand is a string of LEDs.  The two are very different.

You cannot (directly) drive EL (electroluminescent) light strands with a battery box intended for use with a string of LED lights.
EL requires AC to operate.  LEDs require DC, but can operate on AC as long as the peak voltage is high enough and the available current is limited.

The EL controller generates AC from the DC the battery provides.  The required voltage is also rather high - like 100VAC. The EL controller generates high voltage, high frequency, low current power for the strip.

The battery box of a string of LED lights is pretty much just that: a box for the batteries, a connection for the wires, and a switch to turn them on or off.  

So, you can drive a string of LED lights from AC (at least for a while, though it might with time destroy the LEDs) but you can't run EL from DC at all.

If you want to drive EL strands from CR 2032 batteries, you will have to find a controller meant to operate on CR 2032 batteries, or build one yourself.  You might also pull the electronics out of a controller that uses AA cells and try to fit that into the box a of CR 2032 LED battery box - though I don't think there will be enough room in one of them.



Wikipedia on EL.

Adafruit on the practical application of EL.