With the new update, whenever your Pokémon inventory is full (i.e. you have 250 Pokémon with you), and you tap on another Pokémon in the wild to enter into the battle arena to capture, it won't enter into battle. Instead, a popup saying "Your Pokémon Box is full. You can buy more space in the shop." will appear followed by an affirmation selection choice saying OK and a secondary choice saying Go To Upgrades.
So, the exact answer to this question is - NO. You won't be able to catch any Pokémon since it escapes the battle arena by showing a popup alert.
About the situation where the Eggs might hatch after you've reached your inventory; I assume that the egg won't hatch unless you've some free space in your bag to carry the Pokémon.