Sorry, I cannot help, but have the same issue. (fresh install pfSense 2.3.1)
In the Log (Package/SquidGuard/Logs > FilterLog) I see the same "starting, db update done, stopping" behaviour as you described.
I checked from the console to verify whether a separate install of the Squid package via the webGUI is needed. But thats a "no". - When the squidGuard package is installed via the webGUI then it automatically installs also squid 3. (But the webGUI still shows the squid package as not installed. which is simply due to the fact the the squid package in the webGUI is a bundle of several packaged, not squid 3 alone.)
However, I have read that SG somehow looks for squid2, so perhaps that is related to the problem. I will keep looking in the logs for more infos.
EDIT: I am not sure yet as I don't know how to verify it, but it seems that SG is running even thogh its show as stopped. I just had a certain site being the access rejected, and after removing the SG package it worked again. (Direct correlation not verified, but I had the impression.)
Anyone an idea how I can get more reliable infos on the real state of SG?