For smaller leaves, I just collect the whole leaf with the egg and then put a bunch of them together in a plastic dollar-store shoebox lined with a damp paper towel. For the big broad leaved plants, I cut a piece with the egg out, but these seem to dry out faster and so I put those pieces on top of a whole leaf. I don't bring them indoors unless it's pouring rain and there's a chance their box lids will leak at the airholes. I keep the boxes outside, but out of the direct sun. With the air conditioning on it's 15 degrees colder indoors, and I think they do better at the outdoor temperatures.
My cats switch leaf species all the time, they're not very fussy. Whatever gets put in the box is pretty well accepted, but if I had to pick a "favorite" I guess I would say the tropicals.
Certified Monarch Waystation #12
near Philadelphia, Pennsylvania