To my knowledge, or lack thereof, headspace gages won't tell you the difference between 5.56 and .223. Headspace should be the same for both.
On AR systems, the parts (barrel, barrel extension, and bolt) are all held to a standard spec during manufacturing and headspace is not adjustable by mere mortals. IT DOES NOT HURT FOR EASING YOUR MIND TO CHECK IT AFTER YOU ASSEMBLE A NEW RIFLE, but under anywhere near normal circumstances it is not necessary. I have checked headspace with gages on every AR kit I've put together, (like a dozen or so) and never had one out of spec. But I checked it anyway. Standard headspace gages will work, but the military ones are easier. Last place I saw those listed was at Fulton Armory.
Now if you want to determine whether or not you have a TRUE 5.56-spec chamber/throat, here's a link to a tool to check it.
http://www.m-guns.com/tools.php
Click on the 223/556 gage.
On the same page there is a reamer that will bring your chamber to TRUE 5.56 specs, but if you don't have problems when you fire it then why fix anything. But it will add to your education by simply reading about these tools and you will have that knowledge in case of troubles later on in your AR-building career.
By always sticking with a 5.56 chambered barrel, the worries about what type ammo is safe/unsafe to use will be null and void. The 5.56 marked barrel should handle both with no trouble.