What should the inside diameter measurement at the firing end of a muzzle brake for a .223/5.56 NATO rifle?

There is no standard size - it will vary by manufacturer, as they see fit.  I would want at least .020 clearance, which would make for a minimum hole size of .264", but it could easily be .300 or more.

Only way it could be for a 7.62 rifle is if it's .348 or more (since that seems to be what you were worried about from your other post right above this one), as that would be a minimum safe opening for actually firing the larger round.  Even if it's bigger, it doesn't make it a .308 brake - just means it could possibly be used as one, as well.