You can save in the database, or use SharedPreferences, that I believe is the simplest way: 
To make it easier, I recommend creating a class that contains the values I wanted to save: 
class Valores
{
    boolean check1;
    boolean check2;
    boolean check3;
    boolean check4;
}
Method to save this class:
   /**
     * seta os valores no SharedPreferences
     * @param valores
     */
    public void setValores(Valores valores)
    {
        // passamos o nome do objeto que vamos criar e o modo de armazenamento
        final SharedPreferences prefs = getApplicationContext().getSharedPreferences("VALORES_TELA_UM", Context.MODE_PRIVATE);
        final SharedPreferences.Editor editor = prefs.edit();
        // setamos o nome da propriedade e o valor
        editor.putBoolean("CHECK1", valores.check1);
        editor.putBoolean("CHECK2", valores.check2);
        editor.putBoolean("CHECK3", valores.check3);
        editor.putBoolean("CHECK4", valores.check4);
        // commitando as alteracoes
        editor.commit();
    }
Method to redeem values:
  public Valores getValores()
    {
        Valores valores = new Valores();
        final SharedPreferences prefs = getApplicationContext().getSharedPreferences("VALORES_TELA_UM", Context.MODE_PRIVATE);
        valores.check1 = prefs.getBoolean("CHECK1", false);
        valores.check2 = prefs.getBoolean("CHECK2", false);
        valores.check3 = prefs.getBoolean("CHECK3", false);
        valores.check4 = prefs.getBoolean("CHECK4", false);
        return valores;
    }