The core Time::Piece module seems to handle most of your dates but anything earlier than the start of the 20th century can't be parsed and causes an error.

use strict;
use warnings;

use feature qw{ say };
use Time::Piece;

my @dates = (
    q{1960-12-31 23:59:59},
    q{1966-06-24 09:44:00},
    q{1968-12-31 23:59:59},
    q{1969-01-01 00:00:00},
    q{1969-12-31 23:59:59},
    q{1970-01-01 00:00:01},
    q{2000-01-01 00:00:00},
    q{2017-06-24 23:59:59},
    q{2018-06-24 09:44:00},
    q{2238-06-24 09:44:00},
    q{1900-12-31 23:59:59},
    q{1901-01-01 00:00:00},
    q{1900-01-01 00:00:00},
    q{1899-12-31 23:59:59},
    );

foreach my $date ( @dates )
{
    my $tp = Time::Piece->strptime( $date, q{%Y-%m-%d %T} );
    say $date, q{ -> epoch }, $tp->epoch();
}
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The output.

1960-12-31 23:59:59 -> epoch -283996801
1966-06-24 09:44:00 -> epoch -111161760
1968-12-31 23:59:59 -> epoch -31536001
1969-01-01 00:00:00 -> epoch -31536000
1969-12-31 23:59:59 -> epoch -1
1970-01-01 00:00:01 -> epoch 1
2000-01-01 00:00:00 -> epoch 946684800
2017-06-24 23:59:59 -> epoch 1498348799
2018-06-24 09:44:00 -> epoch 1529833440
2238-06-24 09:44:00 -> epoch 8472332640
1900-12-31 23:59:59 -> epoch -2177452801
1901-01-01 00:00:00 -> epoch -2177452800
1900-01-01 00:00:00 -> epoch -2208988800
Error parsing time at /usr/lib/x86_64-linux-gnu/perl/5.22/Time/Piece.p
+m line 469.
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I hope this is of interest.