In James (ICC): A Critical and Exegetical Commentary the author in his text (I have not checked his notes) brings several possible sources for the statement. Amongst them are:
m. Qidd 1.10 (= Mishna Kiddushin 1:10 "and whosoever does not perform a single commandment it shall not be well with him, and he shall not enjoy length of days, and he shall not inherit the Land.")
b Qidd 39b (= Gemoro Kiddushin 39b This is the discussion of the above mishnah. IMHO it does not support the notion that “breaking one commandment means breaking all.”)
Deut 4.2 “Do not add to the word which I command you, nor diminish from it, to observe the commandments of the Lord your God which I command you."
He quotes also Deut 17.20 and Josh 23.6 but they do not seem to me to be relevant.
In summary, there is very little explicit proof from the quoted Jewish sources for the statement.
Further, the Rambam in Hilchos Teshuvo 3:1 says:
“Each and every person has merits and sins. A person whose merits exceed his sins is [termed] righteous.”
If by stumbling on just one point a person was guilty of breaking all of the Torah, a person who had sinned even once could hardly be considered righteous.
So the concept in general is not Jewish; with the exception of those special mitzvos mentioned by Isaac Kotlicky.