I joined the fighters guild, completed the first quest, but for the second quest I can't seem to find where the egg poachers are. Should I be looking for them in one of the houses, or do I need to find the mine, or where can I find them?

The egg mine, called Shulk Egg Mine IIRC, is just southwest of Balmora over the hills. Enter it and make your way to the Queens chamber and you should soon be attacked by the poachers.

Here's a use link to an online Morrowind map including locations like egg mines, tombs and caverns.

Leave Balmora through the town gate (near the fleabus). Continue along the river until you meet two miners (they are bystanders, let your sword be, you can talk with them if you want). The door near them is the entrance to the needed mines.

Thanks! It was the mines I hadn't found yet...