Does it make any difference from a security perspective to use any other IP address ranges than RFC1918 addresses within LAN segments?
No, and it will give you nothing but headaches.
What happens when your email server has to email an MTA that just so happens to have the same IP address as a machine in your subnet? Or when that website your user needs to go to has the same IP address as a machine in your network? Default routes are created saying "If this machine doesn't exist in this network, send it to through the gateway to find said machine", but if that machine is found on your network it will never reach the gateway.
I assume one might think that there is a security benefit by using unconventional IP addressing might deter an attacker footprinting your network, when in actuality if he pops a box he's going to look at the IP address scheme and just scan that network. It won't matter if the network is 10.10.0.0/16 or 184.108.40.206/24.
Private address ranges were invented for a reason and not using them provides no additional security benefits. There is a litany of reasons why "Security through Obscurity" is not a sound security concept, so please save yourself the hassle and just use those in the RFC.