I think that for the index case the situation is not so clear. Both the regex engine and index() will use the same Boyer-Moore routine and for me personally, the regex version is more readable. But as always, YMMV.

use Benchmark qw(:all) ;

my $value = 'FALSE';
my $count = 1_000_000;

cmpthese($count, {
                  'regex' => sub { $value =~ /^true$/i },
                  'eq'    => sub { lc $value eq "true" },
                  'index' => sub { index( lc $value, "true" ) >= 0 },
                 });
[download]

yields

Benchmark: timing 1000000 iterations of eq, index, regex...
        eq:  1 wallclock secs ( 0.89 usr +  0.00 sys =  0.89 CPU) @ 11
+23595.51/s (n=1000000)
     index:  2 wallclock secs ( 1.65 usr +  0.00 sys =  1.65 CPU) @ 60
+6060.61/s (n=1000000)
     regex:  2 wallclock secs ( 1.63 usr +  0.00 sys =  1.63 CPU) @ 61
+3496.93/s (n=1000000)
           Rate index regex    eq
index  606061/s    --   -1%  -46%
regex  613497/s    1%    --  -45%
eq    1123596/s   85%   83%    --
[download]

Update: As AM has pointed out (thank you!), the benchmark above has a bug. Using the tests

                  'regex' => sub { $value =~ /true/i },
                  'regex_anch' => sub { $value =~ /^true$/i },
                  'eq'    => sub { lc $value eq "true" },
                  'index' => sub { index( lc $value, "true" ) >= 0 },
[download]

I get the results

Benchmark: timing 1000000 iterations of eq, index, regex, regex_anch..
+.
        eq:  1 wallclock secs ( 0.88 usr +  0.00 sys =  0.88 CPU) @ 11
+36363.64/s (n=1000000)
     index:  0 wallclock secs ( 1.65 usr +  0.00 sys =  1.65 CPU) @ 60
+6060.61/s (n=1000000)
     regex:  0 wallclock secs ( 1.08 usr +  0.00 sys =  1.08 CPU) @ 92
+5925.93/s (n=1000000)
regex_anch:  2 wallclock secs ( 1.59 usr +  0.00 sys =  1.59 CPU) @ 62
+8930.82/s (n=1000000)
                Rate      index regex_anch      regex         eq
index       606061/s         --        -4%       -35%       -47%
regex_anch  628931/s         4%         --       -32%       -45%
regex       925926/s        53%        47%         --       -19%
eq         1136364/s        87%        81%        23%         --
[download]

with the surprising result that the regex w/o the anchor is faster than the anchored version. Multiple runs yield similar results. As the AM says, one could try many different regex-value combos, but I expect the results to be not far different, precisely because both index and regex engine use the same BM function.