Stylesheet not being called up for certain users

Alright, I have a complicated situation and am mostly looking for a point in the right direction. Here's what is happening.

I have a site @ traveladvicebooks.com that I'm making for a client. It is currently hosted at networksolutions.com. I am using a child theme of the 2013 theme and everything seemed to be going just fine as I developed the site. The client works at a place with really heightened security on their computers and kept insisting that the site was showing up as pure HTML and only the browser's styling were being applied. He uses Windows7 Vista with IE11 at work. Also, he kept insisting that the same symptoms would show up on his apple laptop but only sometimes… I don’t have the browser or OS information about his laptop.

Everyone I showed the site to saw it styled the way the client wanted and I tried every trick that I could think of to replicate the scenario of the site loading with no styling. In addition to the site not working at his place of work, and periodically on his laptop, he kept insisting that some of the people he was showing the site to kept seeing the same issue. The only consistent thing that seemed to be happening was that I saw styling every time I loaded the site and that he didn't every time he loaded the site... At least where he worked. Other than that, it seemed really random. People here and there that he showed the site to would see the site without styling but not once in the multitude of ways I tried could I replicate the scenario and no one I showed the site to had the same issues.

I had to see it with my own eyes so we kind of had to sneak me into his work and, sure enough, the site had no styling. My first attempt in the limited time I had was to shut off all of the plugins, nothing changed. My next attempt was to change themes, still no styling. No matter what I did to alter the site, it still just appeared as pure HTML with no styling. I tried bringing up other sites that I knew were wordpress sites and they had styling. That is, except for some other sites that I had made under a different hosting company local here in Salt Lake City, Utah called xMission.com. I saw an opening.

I had been working with a mutual friend of the client's on another site hosted with host gator so I tried bringing that site up and it had styling. A little more light was shed onto the problem. When my time was basically up at his office, I headed home and decided to try and just re-install his entire site under a sub-directory of the host gator account and see if that worked. Sure enough, when his site was being hosted by Host Gator, all of the styles were applied… Eureka!

My question now becomes this… What could possibly be causing certain users to not be calling up stylesheets while others are calling the stylesheets when the site is loaded? Could it be the PHP version? The Host gator account has PHP 5.4.38 while network solutions is running 5.3.29… Could this possibly be the problem? What would anyone recommend I check or try next to fix this? I’m really stumped at this point…