Select firm, smooth, unwrinkled chayote. Old chayote become very wrinkled and become dry and tough. 
 

Chayote will keep refrigerated for many days

Your first step is to peel off the skin. You’ll either want to wear disposable gloves for this step or peel the chayote under cold running water, as the fruit is known to ooze a slightly sticky substance when peeled. Hold the chayote firm in one hand, and slice the skin off with a common vegetable peeler, rotating the chayote while peeling, as you would a potato.

When the chayote is still green, it is eaten raw, grated into salads. You just have to peel it, and remove the pit.

Chayote that is ripe, though with no sign of sprouting, is boiled or baked;
the tough skin and heart are then removed and the flesh is puréed with milk and butter, as in the French West Indies.

It can be used in gratins, soufflés, beignets, even soups.
 

In Argentina,
They are made into jam.

In Lebanon
It is grown and called French squash (koossa frenjeh). People like to pickle it as it is nice and crunchy.

In the West Indies,
They are cooked like baked potatoes, puréed or used in acras (fritters).
Still in the West Indies, chayote is the principal ingredient in “mange-mêle,” a vegetable stew that includes bacon and coconut milk, which is served as a side dish with spicy foods.

On Reunion Island in the Seychelles,
A veritable chayote cult exists, and there is even a festival devoted to it in Salazie. Here not only the flesh is used (cooked in gratins, fricassees, spicy relishes, even cakes made with dried fruits), but its roots flavour masala and its supple branches are used in making hats.