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
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.