This works for me on RHEL 6.

#!/bin/bash

read -p "Enter a character: " variable
if [ $variable == 'A' ]; then
echo Capital A
else
echo something else
fi

This works for me on RHEL 6.

#!/bin/bash

read -rN1 -p "Enter a character: " variable; echo>&2
if [ "$variable" = 'A' ]; then
  echo Capital A
else
  echo something else
fi

Though if you're going to compare against several characters, you may want to use a case construct instead:

case "$variable" in
  (A)   echo Capital A;;
  (b)   echo Lower case B;;
  ('\') echo Backslash;;
  (*)   echo something else;;
esac