I'm getting errors when I try to run your script with ksh93, mainly because == does not work within [ ... ]. It is unclear to me what shell you're using. You are also executing break as soon as you don't match the pattern, which means you probably only ever run one iteration of your loop. It's unclear what outputs your array, but it is most likely not the code you show.

In any case, the following script does what you want in the ksh93 shell (and in bash as it turns out, and zsh):

array=(
        '1=Napp Slice'
        '4=NNN Issue'
        '3=GG Allow'
        '2=CO-OP'
        '1=Quit'
)

for element in "${array[@]}"; do
        if [[ "$element" == *Quit* ]]; then
                foo=$element
                printf '%s\n' "$foo"
                break
        fi
done

Note the use of [[ ... ]] with the pattern matching operator ==. Also, be careful to quote the expansion of ${array[@]} so that you get each element individually quoted and not split on spaces.

In the comments, you say you want to delete the element from the array. You can't do that if you loop over the elements of the array, but you may unset the element if you instead iterate over the array's indexes.

for index in "${!array[@]}"; do
        if [[ "${array[index]}" == *Quit* ]]; then
                foo=${array[index]}
                unset -v 'array[index]'
                printf '%s\n' "$foo"
                break
        fi
done

Note that this potentially gives you an array with a "missing" element somewhere and that numeric indexes no longer are consecutive.

The code below reproduces approximately what the second code block above does, but for sh, with some extra functionality added. It is provided here purely for our own amusement.

#!/bin/sh

set --  '1=Napp Slice'  \
        '4=NNN Issue'   \
        '3=GG Allow'    \
        '2=CO-OP'       \
        '1=Quit'

unset -v found

for element do
        shift
        case $element in (*Quit*) found=$element; continue; esac
        set -- "$@" "$element"
done

printf 'Element = %s\n' "${found-(was not found)}"
printf 'Remaining list:\n'
printf '\t%s\n' "$@"