Not sure I should answer this, but it is related to Japanese after all, so I'll go ahead.
So, by definition - no. 外食 is a lexeme that consists of more than one stem. That alone is enough to say that it cannot be a morpheme. To elaborate a bit more, both 外 and 食 are actually unbound morphemes (they appear not only as part of larger words), meaning that it's impossible for a combination of the two to be a single morpheme. One might even go as far as say that there are no bound morphemes in Japanese that have a Kanji writing, though I'm not an expert in Japanese linguistics, so take that statement with a grain of salt. An example of a bound morpheme in Japanese, by the way, would be the な modifier (the one you see in 形容動詞).