You're currently just pushing to an array when the checkbox is clicked, no matter whether the click checks or unchecks it. Instead of doing that, I'd suggest toggling a value in array, say $checkbox[$i] for the $i-th checkbox; set it to the checkbox's text when the box is selected, and undef otherwise.

Then when the "Connect" button is clicked and you want to ensure that exactly one checkbox has been selected, use grep to count how many elements of said array are defined.

Here's an example (I've also taken the liberty of formatting your script in a slightly more readable manner):

#!/usr/bin/perl

use strict;
use warnings;
use feature qw/say/;
#use diagnostics;

use Tk;
use Tk::Checkbox;

my $mw = MainWindow->new;
$mw->geometry("150x600");
$mw->title("ConMan");

# open my $FILE, '<', "ESSID", or die "Can't open file: $!";
# my @lines = <$FILE>;

my @lines = <DATA>;

my @checkbox;
my $i = 0;

foreach my $n (@lines) {
    chomp $n;

    my $number = $i++; # private copy for the anonymous sub below
    my $checkButton = $mw->Checkbutton(
        -text       => "$n",
        -onvalue    => 1,
        -offvalue   => 0,
        -command    => sub {
            $checkbox[$number] = defined $checkbox[$number] ? undef : 
+$n;
        },
    )->pack(
        -side       => 'top',
        -anchor     => 'nw'
    );
}

my $connButton = $mw->Button(
    -text       => "Connect",
    -command    => sub {
        my @networks = grep { defined } @checkbox;
        if(@networks == 1) {
            say $networks[0];
        } else {
            say "Select one network, please.";
        }
    }
)->pack(
    -side       => 'left',
    -anchor     => 'sw',
    -padx       => '5',
    -pady       => '5'
);

my $CancelButtons = $mw->Button(
    -text       => "Close",
    -command    => sub {
        say "Closing out ConnMan.";
        $mw->withdraw;
        exit 0;
    }
)->pack(
    -side       => 'right',
    -anchor     => 'se',
    -padx       => '5',
    -pady       => '5'
);

MainLoop;

__DATA__
line1
line2
line3
line4
line5
[download]

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