Is set packing easier when the sets are squares? I am interested in the following problem:

There are n collections of M axis-parallel squares (not necessarily disjoint).
Pick a single square from each collection,
     such that the n selected squares are pairwise interior-disjoint.


One way to approach the problem is to discretize the squares such that each square becomes a set of small squares. This makes the problem an instance of the following set packing problem:

There are n collections of sets (not necessarily disjoint).
Pick a single set from each collection, 
     such that the n selected sets are pairwise disjoint.


The set packing problem is equivalent to the independent set problem, and both are known to be NP-complete.

My question is: does the fact that the sets in this case are discretizations of axis-parallel squares make the problem any easier?

Particularly, is it possible to solve the problem, or at least approximate it to a constant factor, in time polynomial in $n$ (the number of collections), assuming $M$ is constant?
I think your problem is NP-complete already for M=3, as there is a quite straight-forward reduction to it from 3SAT. Just for each variable xi, make a pair of squares, truei and falsei. For a clause Cj, make a pair of squares for each of its literals, e.g., Aj1, Bj1, Aj2, Bj2, Aj3, Bj3. Let Aj1 intersect the respective square of the variable (e.g., truei) (but disjoint from any other similar clause-squares intersecting it), while the Bj1, Bj2 and Bj3 are put somewhere in a disjoint way. Finally, for each clause we have a collection of three squares, which is also Bj1, Bj2 and Bj3.