#Magic mouth could trigger, but alarm will not
General rule: illusions are not creatures and thus will not trigger if the spell requires a creature regardless of creature type
Illusions are visual spell effects that might look, sound, or even act like creatures. But, they are in fact still just magic effects masquerading as creatures.
Jeremy Crawford has confirmed this in a question about detect magic:
Detect magic lets you see an aura only around a visible creature or object, not around an illusion.
And also in this question about Invoke Duplicity:
The illusion of Invoke Duplicity isn't a creature, an enemy, or an ally. It doesn't interact with Sneak Attack.
Note that in these two rulings, Jeremy has said that illusions cannot be affected by abilities or spells that specifically target creatures. Creature is a game term with mechanical implications. And illusions explicitly do not qualify as creatures.
If a spell targets/affects only specifically creaturesonly specifically creatures then you may not use an illusions to qualify.
It also doesn't matter if the spell targets a creature or a creature subtype, it still doesn't work with illusions in general.
Application to specific spells
Moving from the general rule, I'll apply it to the two specific spells you asked about.
- Magic Mouth might get fooled depending on the way the trigger is worded
First, note that magic mouth does not call out creatures anywhere in its spell it will be dependent entirely on how the caster of the spell words their triggers. Since magic mouth depends on auditory or visual triggers it can be fooled depending on how the trigger is worded.
Generally, as long as magic mouth sees or hears something that resembles the creature that you specify in the trigger, it will indeed trigger. As an example: a trigger worded as "anything that looks like a zombie" will definitely trigger off a zombie illusion.
- Alarm cannot be fooled by an illusion
However the mechanics of alarm works, it can detect creatures and creatures only. Alarm does not depend on sight and thus is not fooled by visual illusions. Instead, it does exactly what the description says it does: triggers if a creature enters the area.