Yes. I looked again at the code I snipped from and I don't see any explicit things in it about SSleay. LWP and I presume Mechanize will "just know" about this module and use it if necessary. Of course posts, etc will go to a https URL instead of just http.
You may need a cookie_jar also. And some consideration as to who you are going to "mimic". Here I am pretending to be something pretty stupid. This can affect what the server sends you.
my $ua = LWP::UserAgent->new or die "Problem with the new UserAgent\n"
+;
$ua->cookie_jar(HTTP::Cookies->new);
$ua->agent("Mozilla/4.76 [en] (Windows NT 5.0; U)");
print "And now I'm calling myself ", $ua->agent( ), "!\n";