EEPROM.PUT "er there!"

mattlogue:
I just get 0's.

I haven't messed with these for two decades. I'm not adept.

you are stomping on your 32 bit value moving only one byte in EEPROM.

The EEPROM library can save an object:

#include "EEPROM.h"

constexpr int EEPROM_LOCATION = 0;

struct MyStoredData {
  int a2;
  int a3;
};

void setup() {
  // put your setup code here, to run once:
  Serial.begin(9600);
  pinMode(6, INPUT_PULLUP);
//  pinMode(A2, INPUT);
//  pinMode(A3, INPUT);
}

void loop() {
  // put your main code here, to run repeatedly:
  if (digitalRead(6) == LOW) {
    MyStoredData readings;
    readings.a2 = analogRead(A2);
    readings.a3 = analogRead(A3);
    EEPROM.put(EEPROM_LOCATION, readings);
  }
  if (digitalRead(6) == HIGH) {
    MyStoredData stored;
    EEPROM.get(EEPROM_LOCATION, stored);
    Serial.print("A2: ");
    Serial.print(stored.a2);
    Serial.print("  ...  A3: ");
    Serial.println(stored.a3);
  }
  delay(500);
}