It depends on the actual wording of the Unusual Club

The magic items in the DMG (and elsewhere) have effects described by rule sentences, rather than simple shorthand (as is given in the example). Now, there are different ways to have a magic¹ club deal 2d4 damage:

• Attacks with this club deals 2d4 bludgeoning instead of the normal 1d4.

  Such a wording would mean with shillelagh we'd have two effects trying to replace the 1d4 damage die of the club. In that case the more specific one would apply, which should be shillelagh giving it 1d8 damage.

• Attacks with this club deals an additional 1d4 damage.

  This wording means the clubs magical effect does not interact with shillelagh and the clubs damage with the spell becomes 1d8 + 1d4.

• Attacks with this club deals and additional die of the clubs type.

  (This wording is similar to the bugbear's brute ability) Such a wording would mean the added die is changed when the base die is changed, so the clubs damage with shillelagh becomes 2d8.

If your DM is not providing an actual rule with the weapon description and only the shorthand above, they have introduced a ambiguity into the game and only they can close it/rule on it. If you are the DM, make sure to give your custom magic items proper descriptions, such that your players can better predict the effect of their actions (here casting shillelagh on their special club).

I'm using magic weapon not necessarily as magic for the purpose of game mechanics (such as resistance), but in the sense of a weapon that is a specially awarded (by DM) weapon with rules differentiating it from standard weapons. These are typically magical, and are for design purposes identical with actually magic weapons. Whether the Unusual Club is actually magical doesn't matter to the discussion.

The shillelagh spell does not replace damage if a weapon has multiple damage dice

Note: I agree with the cases in user @Someone_Evil's answer but this is my interpretation for if the weapon actually deals 2d4 damage, without replacing or adding to the usual 1d4:

We know that when the game uses the phrase "damage die" it refers to a single die. This is shown by the fact that getting a brutal criticals on a 2d6 weapon only adds 1d6 to the damage.
Brutal Critical states:

[Y]ou can roll one additional weapon damage die when determining the extra damage for a critical hit...

In order for shillelagh to replace a weapon's damage die with 1d8 the weapon must actually have a damage die, not damage dice. Thus if a weapon has multiple damage dice they are not changed by this spell.

Though the shillelagh spell seems like it would work fully on any club or quarterstaff, the later, more specific clause where the spell relies on a weapon's damage die prevents weapons with damage dice from working as they are not a weapon with a single damage die.

I also do not believe that only one of the weapon's dice will be modified by the spell so that it became a 1d8+1d4 weapon. This is because the spell says it replaces "the weapon's damage die" and not "one of the weapon's damage dice".

Your club will keep its usual 2d4 damage dice while under the effects of the shillelagh spell.

The damage weapon's becomes 1d8 while Shillelagh is active

Your druid would seem to get all or nothing with this spell. Under the idea that spells do what they say they do ...

For the Duration, you can use your Spellcasting Ability instead of Strength for the Attack and Damage Rolls of Melee Attacks using that weapon, and the weapon’s damage die becomes a d8. The weapon also becomes magical, if it isn’t already.

A weapon is listed as having damage, not "damage dice," if you look at the table on PHB p. 149 and see the columns listed. What the shillelagh spell is doing is taking whatever damage die that is listed in the weapons table, and replacing it with 1d8 rather than 1d4, 1d6, 1d8, what have you. Likewise with this weapon. It's damage is determined by rolling 2d4 when it its natural state, but in it's "as changed by this spell for a brief duration" state its damage is determined by rolling a 1d8.

From the above we determine that:

1. The PC can cast shillelagh on a magical weapon.

2. The spell changes the effects of using this weapon as follows:

   • Allow your Spellcasting ability rather than strength for the attack
   • Change the damage dice to 1d8.

   Until the spell ends.

   Congratulations to your druid: they have just made a 2d4 damage die weapon a 1d8 damage die weapon, and they get to use their spellcasting ability for both hit and damage.

   This is the least complicated way to interpret the overlap of the case of this magical club and the spell's effects.

As an aside, I suspect that the devs made sure not to create weapons with 2 dice for a variety of mechanical reasons, part of them being the critical hit mechanic build into the game. In other words, that hypothetical weapon's specs fall outside of the general design framework of how weapons are made in this game.