I wouldn't install it, but unpack it in some work directory and refer to it via LD_LIBRARY_PATH. Downgrading your libc6 package can be quite problematic.

Your customer should be able to supply you with their libc6 version. It might, however, be advisable to use the libc6-dbg package which includes debug symbols. Unfortunately the 7u4 version is an older one and not available anymore.

Your script contains errors as well, for example if you have dos2unix installed your install works but if you don't like I did then it will fail with dependency issues.

I found this by accident as I was making a script file of this to give to my friend who is new to Linux and because I made the scripts on windows I directed him to install it, at the time I did not have dos2unix installed thus I got errors.

here is a copy of the script I made for your solution but have dos2unix installed.

#!/bin/sh
echo "deb http://ftp.debian.org/debian sid main" >> /etc/apt/sources.list
apt-get update
apt-get -t sid install libc6 libc6-dev libc6-dbg
echo "Please remember to hash out sid main from your sources list. /etc/apt/sources.list"

this script has been tested on 3 machines with no errors.

In my situation, the error appears when I try to run an application (compiled on Ubuntu 12.04 LTS) using GLIBC_2.14 on Debian Wheezy (which installs glibc 2.13 by default).

I use a tricky way to run it, and get correct result:

1. Download libc6 and libc6-dev from Ubuntu 12.04 LTS

2. Run dpkg command to install them into a directory (/home/user/fakeroot/ for example):

   $ dpkg -x libc6-dev_2.15-0ubuntu10.6_amd64.deb /home/user/fakeroot/
   $ dpkg -x libc6_2.15-0ubuntu10.6_amd64.deb /home/user/fakeroot/
   

3. Run your command with specified LD_LIBRARY_PATH:

   $ LD_LIBRARY_PATH=/home/user/fakeroot/lib/x86_64-linux-gnu/ YOUR_COMMAND
   

4. My application only uses memcpy() from GLIBC_2.14, and it works.

   I don't know whether it will work successfully for other applications.

Building on Shalini's answer,

#!/bin/bash

SOFTWARE_NAME=$1
SOFTWARE_VERSION=$2

export DOWNLOAD_INSTALL_DIR=$SOFTWARE_NAME-$SOFTWARE_VERSION-download-install
if [ -f $DOWNLOAD_INSTALL_DIR ];
then
    rm -fr $DOWNLOAD_INSTALL_DIR
    echo "Remove $DOWNLOAD_INSTALL_DIR"
fi

mkdir $HOME/$DOWNLOAD_INSTALL_DIR/ && cd $HOME/$DOWNLOAD_INSTALL_DIR

echo "Current directory at $PWD"

export DOWNLOADED_TAR=$HOME/$DOWNLOAD_INSTALL_DIR/$SOFTWARE_NAME-$SOFTWARE_VERSION.tar.gz
if [ ! -f $DOWNLOADED_TAR ]; then
    wget https://ftp.gnu.org/gnu/$SOFTWARE_NAME/$SOFTWARE_NAME-$SOFTWARE_VERSION.tar.gz -P $HOME/$DOWNLOAD_INSTALL_DIR
    echo "Software is downloaded: $SOFTWARE_NAME, version = $SOFTWARE_VERSION "
else
    echo "Software is ALREADY downloaded: $SOFTWARE_NAME, version = $SOFTWARE_VERSION at $DOWNLOADED_TAR"
fi
tar -xvzf $DOWNLOADED_TAR -C $HOME/$DOWNLOAD_INSTALL_DIR
mkdir $HOME/$DOWNLOAD_INSTALL_DIR/build
export SOURCE_DIR=$HOME/$DOWNLOAD_INSTALL_DIR/$SOFTWARE_NAME-$SOFTWARE_VERSION-install
mkdir $SOURCE_DIR
cd $HOME/$DOWNLOAD_INSTALL_DIR/build
~/$DOWNLOAD_INSTALL_DIR/$SOFTWARE_NAME-$SOFTWARE_VERSION/configure --prefix=$SOURCE_DIR
make
make install

export SOFTWARE_PATH=$HOME/$DOWNLOAD_INSTALL_DIR/$SOFTWARE_NAME-$SOFTWARE_VERSION-install/bin/$SOFTWARE_NAME
if [ -f $SOFTWARE_PATH ]; then
    echo "Software is found: $SOFTWARE_NAME, version = $SOFTWARE_VERSION at $SOFTWARE_PATH"
    mv $SOFTWARE_PATH $GRAND_ROOT_BIN
fi

You can use the script like so bash script-name.sh bison 3.8 for downloading GNU's bison with version number 3.8.