Chambers are not the same, with the 5.56 having more headspace and a longer throat for variations in ammo. Watch out on putting military ammo in a .223 as high pressure ammo could be a problem. As Gumaro said, the Wylde chamber is a popular compromise. Good reference is Ammo Oracle:
The AR15.com Ammo Oracle
On the brass, I've always heard that military is thicker and has less case capacity. May really depend on the age of the brass. According to the guys at 6mmbr.com, who are smarter than me (though that's not saying much), most modern brass has about the same capacity. Gumaro's got the link I'm familiar with.
On Sig210:
Very nice pistol, though I prefer the 226 sport.
Best of luck.
Jim