There are a few Egg Collecting quests in Monster Hunter 3 Ultimate, and they can be utterly infuriating if you're not prepared for them. Yep, it's that magic word again: preparation.
Plan Your Route
Egg collection quests commonly block certain routes with rocks, and you can't take cliff-jump transitions when you are carrying an egg. As such you should find the nest and know where you're taking the egg prior to picking it up.
Tag Big Monsters
Bring Paintballs with you (Paintberry + Bomb Casing) so that you can mark large monsters for your map.
If you know where the huge monster is then it can inform you as to how you should act – a large monster's entry roar can cause you to drop an egg if you've not taken the precautions to avoid such flinching through armour or food buffs.
Bring Dash Juice
You'll need to run with the egg, constantly. As such, Dash Juice is a key assistant in helping make this quest type much less painful. A Dash Juice is found by combining a Rare Steak with a Catalyst, while Mega Dash Juice is made through a combination of Dash Juice and Well-done Steak
You will encounter enemies, this is an unavoidable truth. The game will often fill a map with aggressive creatures upon grabbing an egg. It's important, therefore, to walk widely around these beasts to avoid any sort of sudden attack.
Don't walk too far around them, you don't want to waste time and let a big monster arrive, but don't rub shoulders with them either.