Is it decidable whether a linear language contains a square?
It is indeed undecidable whether a linear grammar generates a square. Same technique, reduction from PCP.

Starting with the two sequences of strings  $(u_1,\dots,u_n)$, $(v_1,\dots,v_n)$  that define the PCP we can build a linear grammar with strings of the form
$u_{i_1}\dots u_{i_k} j_k \dots j_1 \# v_{j_1}\dots v_{j_k} i_k \dots i_1 \# $.

Rules like $S\to A\#$; 

$A \to u_i A i \mid u_i B i$; 

$B \to j A v_j \mid j v_j$.

Such a string is a square if both the indices and the strings match, hence iff we generate a solution of PCP.

(I really did not know the answer when I posted the question... Sorry. I started examining ways to show decidability.)