"egg" zuchini
 

I am planting French egg zuchini. The vegetable is the size of an egg. Do these have to staked or put on a trellis like cucumbers do?

No staking, they have no tendrils to hold on with.

I have not grown the variety that you are growing, but I have never done any staking for eggplants. They grow like a small stocky determinate tomato and always stand upright like a bush even when they are loaded with fruit.