When they type quit, they actually type "enter" after the four letters; that becomes part of the string. When the string is converted to a number (for the == compares) the trailing newline (like all whitespace) is silently ignored, but eq is tripped up by it.

Traditionally, one handles this by saying:

while (my $guess = <STDIN>) {
   chomp($guess);
   ...
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See chomp.