Firstly, there's nothing special about the private IP ranges; NAT will work just as well on any address, it's just that there's really no need for it with publicly-routable addresses.
For a proxy to use with MikroTik Router OS, I'd suggest Squid but probably any HTTP proxy server would work; if you're in a Microsoft environment you may prefer their proxy offering.
In terms of 'hiding' internal IPs, what I believe you're referring to is the HTTP X-Forwarded-For
header. You should be able to configure most proxy servers to omit this header (it's not part of the standard anyway, hence the X-
prefix). I'd strongly recommend testing this before deploying to production by connecting through the new proxy to a server you control and examining the HTTP headers the server receives.