I'm playing Pokemon Omega Ruby and hatched a Meowth. The Pokemon glittered when it came out of the egg, and it had a sort of yellow background with orange dots, but the Meowth is definitely not shiny. I haven't found any indication of what that was about by searching online. Can somebody point out what it is that I'm missing here?
That's completely normal. The background and hatch effect was changed a bit from XY's but every egg in ORAS (even shiny ones) will hatch with that yellow and orange polkadot background and have sparkly effects.
The only way to tell if your newly hatched pokemon is shiny is to look at its colors, or just wait and look for the red star on its summary screen later.