First of all, with regards to your hash %squares, I suggest you make it easier to read by making use of the => operator which acts just like a comma, but has the benefit of clarity in this case:

%squares = (
    a1 => $a1, 
    a2 => $a2, 
    a3 => $a3, 
    b1 => $b1
    b2 => $b2
    b3 => $b3
    c1 => $c1
    c2 => $c2
    c3 => $c3
);

As to why your hash value is not being "emptied", in make_blank() what you are doing is concatenating nothing onto the value of the chosen square:

$squares{$key} .= "";

The .= is the concatenation operator. You want to use a regular assignment:

$squares{$key} = "";

Or just undef() it:

undef($squares{$key});

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Amel
This is my cool %SIG