I decided to go out to eat.
I told my son.
"Tell mummy we are going out to eat."
"Tell mummy we are eating out."
Should I use outside or out in the context above?
Your question relates to the following in the context of "we are going out(side) to eat".
Out vs outside
Some definitions:
out adverb 3.1 In or to a public place for purposes of pleasure or entertainment. ‘an evening out at a restaurant’ - ODO
outside preposition & adverb Situated or moving beyond the confines or boundaries of. ‘I stepped outside the marquee for a breather’ - ODO
*Going out to eat / eating out suggests going to a restaurant, whereas going outside to eat / eating outside suggests remaining at the property but not in the dwelling itself (e.g. eating on the balcony).
First sentence: use out because you are using out as a part of phrasal verb not preposition.
Second sentence: You can use both but I usually say out because it's shorter.