The question is stated in the title of this post. 

It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,z^n$ as rational functions of $z$, but the expressions seem hard to analyze. 

Letting 
$$d_n:=(-1)^{n (n - 1)/2} (-1)^{n - 1}\,\text{discriminant}(p_n),$$
we get 
$$(d_1,\dots,d_{15})=(-4, 1, 44, 279, 2300, 57425, 841436, 14201719, 442321436, 
10095992037, 254419300556, 9827983382723, 304507125159644, 
10182574354147897, 472932455198902268).$$

There is [nothing in the OEIS][1] on 
$$4, 1, 44, 279, 2300, 57425, 841436, 14201719$$ 
or on 
$$1, 44, 279, 2300, 57425, 841436, 14201719.$$


  [1]: https://oeis.org/search?q=1%2C%2044%2C%20279%2C%202300%2C%2057425%2C%20841436%2C%2014201719&language=english&go=Search