Can the extra action from the haste spell be used by a tortle (from The Tortle Package) to withdraw into their shell?
No, they can't - but they can just use their regular action to do that instead
You can withdraw into your shell as an action. Until you emerge, you gain a +4 bonus to AC, and you have advantage on Strength and Constitution saving throws. While in your shell, you are prone, your speed is 0 and can’t increase, you have disadvantage on Dexterity saving throws, you can’t take reactions, and the only action you can take is a bonus action to emerge from your shell.
If you're playing a tortle, you can do this as an action (as long as you haven't already withdrawn into your shell, of course).
The haste spell description says:
Choose a willing creature that you can see within range. Until the spell ends, the target's speed is doubled, it gains a +2 bonus to AC, it has advantage on Dexterity saving throws, and it gains an additional action on each of its turns. That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action.
When the spell ends, the target can't move or take actions until after its next turn, as a wave of lethargy sweeps over it.
As you can see, the spell description specifically lays out what you can do with the extra action the spell gives you - and using your racial trait is not one of those options.
However, this is easily circumvented by simply using the action you already have normally to withdraw into your shell. (If you're the caster, you can't do this on the turn you initially cast the spell, since obviously you've already used your regular action to cast the spell.) Then your additional action from the spell is limited to the 5 options listed in the description.
Generally, this is only really a problem when you want to take two actions that aren't listed in the spell description (e.g. the Cast a Spell action, and the Dodge action) on the same turn. Otherwise, if you're okay with using one of your two actions to take an action listed in the description of haste, then you can use your regular action to do whatever else and your additional action from haste to take one of the specified action options.
This is generally fairly straightforward, because nothing about the haste spell constrains the timing of the extra action, or the order in which you must take your multiple actions. (There are no general rules constraining the order in which you can take actions, not including bonus actions with a specified timing.) As a result, you can take your regular action and then your extra action from haste, or vice versa.