So basically what you are saying is you are fucking off at work, someone said something to you about it, and your own paranoia set in. So you figure you will embrace your own IT nerdiness throw the tinfoil hat umbrella out here in the hopes of stirring enough black helicopter theorists to get someone to give a shit. Does that about sum it up?
If you've been in IT for 15 years and know anything about security and/or data detection then you know that us having HTTPS isn't going to stop someone from being able to mine information via data packet sniffing. Because the web host addresses and port numbers are part of the underlying TCP/IP protocols, then HTTPS isn't going to protect their disclosure. So even IF everything were correctly configured on the web server your typical 15 year old that has spent a half hour googling on how to hack someone's password on an internet forum board is going to have enough basic knowledge to infer the IP address and port number of the web server and quite possibly the domain name that someone is communicating with as well as the amount of data transfer and duration of the communication.
So if you are trying to keep your employer's web police from knowing what you are doing.. us having HTTPS isn't going to do it. Seems like an easier way to hide your tracks would be for you to set up VPN to your home router, remote desktop to your home PC, then do your "non work related" web surfing from there.