Several small problems are tripping you:

&loginpage if $loggedin = 0;

You mean to compare with ==, not assign with = here. loginpage() will never be called because the assignment expression evaluates to 0, or false.

open (FILE, "users.dat");

You're not checking to see if the open succeeds. Getting the permissions right on a data file can be tricky, so it's worth dying or warning if you can't open the file.

if (&checklogin($userdef, $passdef) == 0) {

This is the second time you're calling this function. It's reasonable to assume you only need to call it once per invocation.

print header,
            start_html('Login Page'),
            hr;
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This should be a comma. The rest of the HTML generating functions do nothing in void context and won't be printed to the screen.

if (param()) {
      $userdef = $query->param('user');
      $passdef = $query->param('pass');
}
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You're mixing the functional and object oriented interfaces to CGI here. Worse, I don't see where you've defined $query, so this is effectively useless.

If you turned on warnings with -w or the warnings module and enabled strict, Perl would let you know about most of these errors. Just run it from the command line:

perl -wc -Mdiagnostics mycgi.pl

You'll have a lot of things to fix, but if you're having trouble debugging, you need all the information you can get. This is how to let Perl help you.