As requested by Franz, here is the short Magma code looking for solutions in cyclotomic units:

test_n := function(n)
K<z> := CyclotomicField(n);
O := MaximalOrder(K);
I := ideal<O|4>;
R := quo<O|I>;
G,p := MultiplicativeGroup(R);
p1 := p^-1;
H := sub<G|[p1(z)] cat [p1(c) : c in CyclotomicUnits(K)]>;
return [p : p in PrimeDivisors(n) | p ne 2 and p1(1-z^(n div p)) in H];
end function;


I have run this for $n$ up to $171$, always returning an empty set. This is naive code and can be speeded up like Franz says.

Given David's answer, it is now interesting to look at the even case. Here there are in fact many solutions. For $n\le 100$ there are solutions when $n$ is 28, 56, 60, 92, with $p$ being 7, 7, 3, 23, respectively.

1

As requested by Franz, here is the short Magma code looking for solutions in cyclotomic units:

test_n := function(n)
K<z> := CyclotomicField(n);
O := MaximalOrder(K);
I := ideal<O|4>;
R := quo<O|I>;
G,p := MultiplicativeGroup(R);
p1 := p^-1;
H := sub<G|[p1(z)] cat [p1(c) : c in CyclotomicUnits(K)]>;
return [p : p in PrimeDivisors(n) | p1(1-z^(n div p)) in H];
end function;


I have run this for $n$ up to $171$, always returning an empty set. This is naive code and can be speeded up like Franz says.