Sure you can do that, you need to stay at least 3 months before asking for it first.
Note that how much you get is barely enough to live (but it fulfills its goal which it to reduce poverty).
(and there's a good chance the program will be scraped at some point, since while it fulfilled its goal with reducing poverty, it didn't have much impact in term of employment and there is still a ceiling effect even so it was supposed to remove them).
(and I'm not sure which part of reducing poverty doesn't make much sense, and as a EU citizen you can profit from any wellfare system within the EU)
As an EU citizen can't I just relocate there and collect without ever having contributed to french social security?
Doesn't seem to make much sense...