There's been a significant change in the latest OS X update involving the way Google Chrome (53) now handles SSL certificates: now an intermediate certificate file is required.
This problem doesn't affect Chrome Canary.

*The fix*

cat primary_cert.crt intermediate_cert.crt >> bundle.crt

*Apache*

<VirtualHost *:443>
ServerName site.com
ServerAlias www.site.com

DocumentRoot /home/site/www

SSLEngine on
SSLCertificateFile /home/site/server.crt
SSLCertificateKeyFile /home/site/server.key
SSLCertificateChainFile /home/site/bundle.crt
</VirtualHost>

*nginx*

server {
    server_name  site.com www.site.com;
    listen 443 default_server ssl;
    root /home/site/www;
    index index.html index.php;

    ssl_certificate /home/site/bundle.crt;
    ssl_certificate_key /home/site/server.key;
    ssl on;
}

*Node.js*

'use strict';

var https = require('https');
var express = require('express');
var app = express();

var sslOptions = {
    key: fs.readFileSync('server.key'),
    cert: fs.readFileSync('server.crt'),
    ca: fs.readFileSync('bundle.crt')
};

https.createServer(sslOptions, app).listen(3000);

Hope this helps.