I'm baking cupcakes and I want to frost the cupcakes the night before I give them to my friend the next day. Can I put the buttercream in the fridge? What will happen to it? If I make a buttercream with eggs, can I still frost in advance and put it in the refrigerator?

I don't know of any buttercream recipes that will not refrigerate. Unless you have a really funky recipe, I'd say go for it.

What do you mean a bc recipe with eggs, do you mean a cooked bc like SMBC/IMBC? I always refrigerate my IMBC cakes anyway - I just allow at least an hour out of the fridge for the icing to soften before eating.