Let $f(z)=f(z+a)=f(z+b),\ a,b\in\mathbb C$ be a not constant meromorphic function, which is periodic and let $a,b$ be lineare independent. Show that $f$ has no zeros or singularities on the boundary of $$G_u=\{u+\lambda a+\mu b|\lambda,\mu\in[0,1)\}$$ for some $u\in\mathbb C$.
How do I know that, without knowing an explicit formula? Can I take a random $f(z)$?
No. The question should have been phrased as follows: "Given any nonzero meromorphic function $f$ and linearly independent complex $a,b$ such that $f(z) = f(z+a) = f(z+b)$ for every complex $z$, prove that $f$ has no ... ". So you must, given any such $f$, which you cannot choose, find some $u$ with the desired properties.
The "nonzero" condition is important, and the question is false without it.
And this should not be hard. After all, there are uncountably many possible pairs of vertical lines at positions $(x,x+1)$ where $x \in [0,\frac12]$, but only countably many zeroes or singularities, so some pair has no zero or singularity on them. Similarly we can find a pair of horizontal lines that are a unit apart and have no zero or singularity on them. Combining gives the square we want.