It's a theorem of Stallings and Swan that a group of cohomological dimension one is free.
By a theorem of Serre, torsion-free groups and their finite index subgroups have the same cohomological dimension.
So, a torsion-free group is free if and only if its finite index subgroups are free.
(Here are the references. For Stallings-Swan, see
John R. Stallings, "On torsion-free groups with infinitely many ends", Annals of Mathematics 88 (1968), 312–334.
Richard G. Swan, "Groups of cohomological dimension one", Journal of Algebra 12 (1969), 585–610.
Serre's theorem is in Brown's book "Cohomology of Groups.")