Forgetting the squarefree condition for a moment, the number of integers up to $N$ that are not divisible by any primes less than $N^{1/k}$ is asymptotic to
$$
\omega(k) \frac N{\log N} \sim e^\gamma \omega(k) N \prod_{p\le N^{1/k}} \bigg( 1-\frac1p \bigg),
$$
where $\omega$ is the Buchstab function. (In particular, the first half of the density you derive in your answer is not correct: there is a correction factor of the form $e^\gamma \omega(k)$.)
Now the number of integers up to $N$ that are divisible by the square of a particular prime $p$ is at most $N/p^2$. So the number of integers up to $N$ that are divisible by the square of a prime greater than $N^{1/k}$ is at most
$$
\sum_{p>N^{1/k}} \frac N{p^2} < N \sum_{n>N^{1/k}} \frac1{n^2} < N \cdot \frac1{N^{1/k}}.
$$
Therefore the above asymptotic also holds for squarefree numbers not divisible by small primes.
Here is an elaboration of my comment to the question. Suppose that the primes are (pre)-periodic $\pmod k$ (with $k>2$). Let $\chi$ be a non-trivial character $\pmod k$. Then the assumption on the primes implies that
$$
S(x,\chi) = \sum_{p\le x} \chi(p),
$$
is bounded.
Now consider
$$
\log L(s,\chi) = \sum_{j=1}^{\infty} \frac{1}{j} \sum_{p} \frac{\chi(p)^j}{p^{js}}.
$$
A priori this is well defined and analytic in Re$(s)>1$. The terms with $j\ge 2$ are analytic in Re$(s)>1/2$. The assumption on the periodicity of the primes gives that
$$
\sum_{p} \frac{\chi(p)}{p^s} = \int_{1}^{\infty} S(x,\chi) \frac{s}{x^{s+1}} dx,
$$
is analytic in Re$(s)>0$. Thus we conclude that $\log L(s,\chi)$ is analytic in Re$(s)>1/2$ (this is the Riemann Hypothesis, but of course a false assumption implies many things!).
Now suppose that $\chi$ is a non-trivial quadratic character. Then from above we see that for Re$(s)>1/2$ we have
$$
\log L(s,\chi) = O(1)+ \frac{1}{2} \sum_{p} \frac{\chi(p)^2}{p^{2s}} = O(1) + \frac{1}{2} \sum_{p} \frac{1}{p^{2s}},
$$
since $\chi$ is quadratic.
If we let $s$ take real values tending to $1/2$ from above, this gives a contradiction (as $L(1/2,\chi)$ is bounded, whereas exponentiating the above would imply that it goes to infinity).
Alternatively take $\chi$ to be a complex character (this works for $k\neq 3, 4, 6, 8$) and then the prime square terms above are also bounded (for Re$(s)>0$) and we conclude that $\log L(s,\chi)$ is analytic in Re$(s)>1/3$. Hence the $L$-function is non-zero in that region, and by the functional equation (say $\chi$ is primitive) we conclude that the $L$-function can only have trivial zeros. This is a contradiction.
Thirdly, as noted before Daniel Shiu's result on Strings of Congruent primes (JLMS 2000) also gives the result (although this is harder than the proof above). Nowadays the Maynard(-Tao) machinery gives relatively easy access to such results (and in a stronger form).
Finally, as Terry Tao observes in the comments, Gowers's question on why the Riemann zeta function has zeros is certainly relevant here, and see also my answer to that question. In that spirit, one may say that the primes are not periodic $\pmod k$, because the integers are!
Best Answer
Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107; in fact he gives an explicit constant here that he says he cannot improve), and page 8 of http://hsb.org.hu/~p_erdos/1981-21.pdf .
A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .