Can't get Facebook or LinkedIn login working

Copy-paste following url in the “Website URL” and “Javascript API Domains” options of LinkedIn app (see step 6 and 11 at https://thechamplord.wordpress.com/2014/01/26/getting-linkedin-api-key/) you configured for Social Login and save the changes:
https://www.travelnasia.com

Facebook icon should appear after fixing LinkedIn issue.

That worked for LinkedIn. But still no Facebook login icon. Did I need to make a similar change in the FB settings?

To resolve Facebook issue;
Open “wp-content/plugins/super-socializer/css/front.css”, search following code (at line number 284)

.theChampFacebookButton {
	background-position: -0px -0px;
	width: 32px;
	height: 32px;
	display: none;
}

Replace above with following:

.theChampFacebookButton {
	background-position: -0px -0px;
	width: 32px;
	height: 32px;
	display: block;
}

Save the file back.

That worked. Thanks. I assume you’ll roll that into any future upgrades to the plugin?

Yes, definitely.