Who wrote Shalom Aleichem and Eishes Chayil? Is it minhag Yisrael?
As Isaac pointed out, Eishes Chayil itself is a Biblical passage.
The practice of saying both of these on Friday night is found in Siddur Shaar Hashamayim (by the Shelah Hakadosh, R' Yeshayah Halevi Horowitz), first printed in 1717. I don't know whether there are earlier printed sources for this custom.
Shalom Aleichem was written in the 17th century by Mekubalim the source from wikpedia According to the Talmud, two angels accompany people on their way back home from synagogue on Friday night — a good angel and a evil angel. If the house has been prepared for the Shabbat ("the lamp has been lit, the table set, and his couch spread"), the good angel gives a blessing that the next Shabbat will be the same, and the evil angel is forced to respond "Amen". but if the home is not prepared for Shabbat, the evil angel wishes that the next Shabbat will be the same, and the good angel is forced to respond "Amen"
It has become the minhag for many of Ashkenazic background, but to say a custom started in the past 500 years is a Minhag for all Yisrael you would need to prove that no Torah observant Jew doesn't have the minhag to say them.
What I mean is that I am unaware of anyone (who is observant and has a tradition) that specifically does not say them. If everyone, all Ashkenazim, Sfardim, Teimanim, Parsim, Kurdistanim, Roma'im, etc. each tradition, if all of their entire group has the minhag to say Sholom Aleichem and Eishes Chayil then it clearly is Minhag Yisrael.
But if you find one group that there are those within it who do not have the minhag then for them it cannot be considered minhag yisrael.
Rambam in his preface to his Commentary on the Mishna says that one who does not keep Minhag Yisrael, on him the verse is applied "One who breaks down a fence will be bit by a snake." There is much to consider regarding the implications of this statement. Does he mean a universally-Jewish minhag? Or does he mean any minhag that has valid roots in our tradition and that one's forebears have kept? Or does he mean the valid minhag of the community one belongs to?
There is much written on this subject and I will need to do some review and some more reseach to present it properly.